Értékelések
2005 · 24 oldal (290 KB)
magyar
46
2007. június 21.
Bejelentkezés szükséges
Beágyazás
Nincs még értékelés. Legyél Te az első!
Tartalmi kivonat
!!" ! " # $ " " ! " # $ % & Rd $ K # ' ϕ1 , ϕ2 , . , ϕr (r ≥ 2) ' K = ϕ1 (K) ∪ ∪ ϕr (K) ($) *+, ϕ1 , ϕ2 , . , ϕr )'
# K K = ϕ1 (K) ∪ . ∪ ϕr (K) - $ $' ! K ' . / $ ' 0 ϕ1 , ϕ2 , , ϕr ' K = ϕ1 (K) . ϕr (K) K ' # $ $' & . K ' ' {ϕ1 , ϕ2 , . , ϕr } K = ϕ1 (K) ϕr (K) & K . ' (ϕi1 ◦ ◦ ϕin )(K) ! . ' 1 r p1 , . pr ∈ (0, 1) μ ' 2 3 i=1 pi = 1 % K μ◦ϕi = pi ·μ (1 ≤ i ≤ r) H ⊂ K 3 μ(ϕi (H)) = pi · μ(H) (1 ≤ i ≤ r) # $ *+, 4 5 6
& . $ ' / 7 0 $ ' 1 $ ($ 8 # ' $ / # $ *4, 5, 9, :,0 . # 2 /0 & $ / $ 0 7 # # 2 7 $ /7 0 *9, % ' $; K ' $ ' μ g ' K ∩ g(K) K μ K ∩ g(K) = μ K (K ∩ g(K)) 5 < !
. / 0 ; " $ ' ' K ' μ c < 1 g ' μ K ∩ g(K) < c K ⊂ g(K) 5 = >? K = ξ1 (K) ∪ . ∪ ξr (K) # $ ξi ? / ? . )' # )' 0 . ? #) ? 3 . *<, ")"$ @, #A " B ($ 8 ' C $ 1 *+D, $ 2 ($ 8 $ E *+4, $ . + ($ 8 *++, # !
$ ' - . ! " #A ? ' # 2 /! #$ #A $ 0 F ' # 2 G H I *J, $ # $ & 1 $ $ *=, 1 K ⊂ Rd 3 μ K 3 & μ ' $ H ⊂ K 3 g ' g(H) ⊂ K μ(H) = μ(g(H)) ( ' ' $ /4 9 6 0;
μ∗ ' 3 Rd μ∗ |K = μ ! / ' ?0 ' K # 2 K . ' & . ? $ $ ! $ ' ' ' )' $ ' & . $ " $ ' ' K ⊂ Rd ' μH ' 3 Rd μH (K) = 1 . ' ($ 8 / # $ *+5, +D= 0 & ($ 8 ' ' '
μH K ' $ ' ' ' ! ($ 8 $$ 5 = ' )' F μ ' ' 2 " $ $ . ' /9 9 0 ! μ pi ' ' ' ϕi ' ' 2 ' μ ' & ' . . ' 9 : 4 !
$ # . L)' ! $ ' ' ' . # $ {ϕ1 , ϕ2 , . , ϕr } {ϕ1 ◦ ϕ1 , ϕ1 ◦ ϕ2 , , ϕ1 ◦ ϕr , ϕ2 , ϕ3 , , ϕr } & K {ϕ1 , . , ϕr } ' (ϕi1 ◦ ◦ ϕin )(K) (ϕi1 ◦. ◦ϕin )(K) )' F $ M ' ; M $ M I = (i1 , i2 , . , in ) # $ ϕI = ϕi1 ◦ ◦ ϕin pI = pi1 pi2 pin & I = ∅ $ M ϕ∅ = p∅ = 1 N K ' ϕI (K) ' h ' h(K) ' h(ϕI (K)) %& !' K K
" ! # $ μ ◦ ϕi = pi · μ μ(K) = 1 # μ p1 , . , pr ∈ (0, 1) {ϕ1 , ϕ2 , . , ϕr } # r i=1 pi = 1 (i = 1, . , r) 3 . $ σ # 3 ' 1$ ' ) 2 . O P . ) & $ F = {ϕ1 , ϕ2 , . , ϕr } L F1 = {ψ1 , ψ2 , . , ψs } 1 ≤ i ≤ r I $ M ϕi = ψI ! L . F1 ≤ F F L F1 ! . . ' L ! . L F ' / $ '0 . /
0 ) . / : 0 & . ) ' ' ' F . L %& !' K ⊂ Rd ' % " μ μ # &# g # g(H) ⊂ K K " & H ⊂ K " μ(H) = μ(g(H)) ! K = Rd L)' ' ) & 4 9 6 $ L)'; K ' Rd ' 5 - αi ϕi ' F # ' K ' # F ϕI (K) ϕJ (K) ' αI = αJ & 9 9
)' μ # ' αi1 αi2 . αin = αj1 αj2 αjm pi1 pi2 . pin = pj1 pj2 pjm μ pi " $ ' ' ' . ' ($ 8 # *+5, $ + / # ($ 8 ' 0 #$ # ' & ($ 8 . pi = αsi s # ' αsi = 1 ## ' ' & Rd $ ' ' ' # ' " ' x αAx + b α ≥ 0 ' A n × n M ' b ∈ Rd # ! 2 ' Rd +d
; d2 αA 2 M ' d # b ' F Rd +d 2 $ ' Rd +d 2 H ' $ " ' Rd +d H $ # # # & ' # )' $ ' # ) # # )' ! ) # # $ ) # L # ' /# ) # 0 (h, g) h ◦ g g g−1 . # ! 2 ' gn ∈ H ' # h ∈ H ' /gn h0 K ⊂ Rd # ε > 0 N n ≥ N x ∈ K gn (x), h(x) < ε %
dist # ( ! K = ϕ1 (K) . ϕr (K) ! 0 ≤ p1 , . , pr ≤ 1 " K & pi = 1 H ⊂ K " μ # $ 1 ≤ i ≤ r & μ(ϕi (H)) = pi · μ(H) # μ ◦ ϕi = pi · μ " ! F ϕI (K) ; μk (H) = inf pI ! L $ μk ∞ pI i : H ⊂ i=1 ∞ i=1 ϕIi (K) . & K $ $ ' $ ' # 3 $ 3 σ # μ 9 & $ μ(K) = 1 F μ(K) ≤ 1 F K .
' K ⊂ ∞ # i=1 ϕIi (K) & ϕIi (K) ) K . K F $ . ! M K = ni=1 ϕIi (K) ϕIi (K) ∩ ϕIj (K) = ∅ i = j ! . # $ M ' $)' # $ ni=1 pIi = 1 μ(K) = 1 & μk L)' # H ⊂ K 3 μ(ϕi (H)) = pi · μ(H) 1 ≤ i ≤ r 1 μ ' 2 3 μ ◦ ϕi = pi · μ (1 ≤ i ≤ r) & μ μ μ (ϕI ) = μ(ϕI ) = pI K ' $ $' μ μ μk L)' # μ(H) = inf {μ(G) : H ⊂ G } , μ (H) ≤ μ (G) = μ(G)
(H ⊂ G ). ! μ (H) ≤ μ(H) $ μ (K H) ≤ μ(K H) #$ H ⊂ K 3 μ (H) = μ(H) ( ! ∅ = K ⊂ Rd " μ∗ # &# " K & μ # &# " Rd & μ∗ |K = μ " ! ( H ⊂ Rd 3 . K ' ' # μ∗ ( . $ H = i Hi ' 3 #)' Hi ⊂ gi (K) gi ' 1 μ∗ (H) = μ(gi−1 (Hi )). i & $ ' L)' . #)' ' ' 1 H = j Hj Hj ⊂ gj (K) #)' H = i,j Hi ∩ Hj #)'; i μ(gi−1 (Hi )) = i μ(gi−1 (Hi ∩ Hj )) = j
= i j μ(gj−1 (Hi ∩ Hj )) = μ(gj−1 (Hj )). j F μ∗ . #)' ' ' 3 $ μ∗ μ∗ (∅) = 0 & σ ' $ Ai 3 ( . K ' ' # $' $ $' # Ai = j Hij Hij ⊂ gij (K) ! i Ai = i,j Hij #)'' ' μ∗ i Ai = i μ∗ (Ai ) : & # μ∗ |K = μ μ∗ ' " $ # ) K = ϕ1 (K) . ϕr (K) μ ε > 0& a(K) " ! " H . H ⊂ K # $ "
μ H ∩ a(K) > (1 − ε)μ a(K) ( # H (1 + ε) μ(H) > ' i ϕIi (K) μ(ϕIi (K)). i ( i (1 + ε) μ(H ∩ ϕIi (K)) > μ(ϕIi (K)), ϕIi . a ! (1+ε) μ H ∩ϕIi (K) ≤ μ(ϕIi (K)) (1 + ε) μ(H) = (1 + ε) μ H ∩ ϕIi (K) ≤ μ ϕi (K) , i . * ) K = ϕ1 (K) . ϕr (K) H i ai (K) μ H ⊂ K " ε > 0& μ H ∩ai (K) > (1−ε)μ ai (K) ( ai (K) $ " ! " ' ' $ ai (K) & n # /n = 0, 1, 2, . 0 $ / 20 an (K) an
(K) ∩ ai (K) = ∅ (0 ≤ i < n) μ H ∩ an (K) > (1 − ε)μ an (K) ( ) $ $ & $ H i ai (K) $ 2 . # 2 ! 4 : 1 a(K) μ (H i ai (K)) ∩ a(K) > # & (1 − ε)μ a(K) a(K) μ H ∩ a(K) > (1 − ε)μ a(K) ( # $ a(K) a(K) ) # ai (K) ( # a(K) $) $ ai (K) Q ai (K) ⊃ a(K) ai (K) ⊂ a(K) + ) K ⊂ Rd μ=K μ # $ " Rd & gn # gn h K) 1 !
μ(h(K) ∩ K) = 1 K ⊂ h(K) J μ(gn (K) ∩ " ! . μ(h(K) ∩ K) = q < 1 - h(K)ε h(K) ε $ " ∞ μ h(K)1/n ∩ n=1 (h(K)1/n ∩ K) = h(K) ∩ K μ K μ h(K) ∩ K = q 1 ε > 0 μ h(K)ε ∩ K ≤ 1+q 2 < 1 ! 1+q n gn (K) ⊂ h(K)ε μ gn (K) ∩ K ≤ 2 μ(gn (K) ∩ K) 1 μ(h(K) ∩ K) = 1 " μ ' K K # 2 & K h(K) $ 2 K ⊂ h(K) ) K = ϕ1 (K) . ϕr (K) μ K & ! A ) μ(A ∩ K) = 0 # A⊃K " ! " J $ M ϕJ (K) ? ## ' K ϕJ ' &
$' ϕJ (K) ⊂ K ' ! A ? A ⊃ K # A ⊃ ϕJ (K) F $ M ? . # K x1 , x2 , , xk 1 U1 , U2 , . , Uk M # # k # ? . " 1 ≤ i ≤ k ϕIi (K) Ui 1 ε = min1≤i≤k μ(ϕIi (K)) > 0 F K ' A ? ϕIi (K) K $ J $ M A ? (ϕJ ◦ ϕIi )(K) 1 ≤ i ≤ k A ⊃ ϕJ (K) A ⊃ K ! μ(A ∩ ϕJ (K)) ≤ (1 − ε)μ(ϕJ (K)) A ⊃ K . A ⊃ K μ(A ∩ K) > 0 & 4 : 1 ϕJ (K) (1 − ε)μ(ϕJ (K)) < μ(A ∩ ϕJ (K)), . &
. #$ $ K ' K g(K) ' # /g ' 0 E $ K ⊂ g(K) & 4 < 6 g ' K A ? μ(g(K) ∩ K) = 0 K ⊂ g(A) ! ) ' . $ A LM ' ) A ' ' $ ' ? $ ' H " ## ' . K ? $ $ ! . ' . =
! . $; " $ ' ' K ' μ c < 1 g ' μ(g(K) ∩ K) < c K ⊂ g(K) & ) ' #' $ . 3 $ ' G # g ∈ H g ∈ G g (K) ∩ K = g(K) ∩ K 2 G ' $ V, U ⊂ H V ⊂ U ⊂ H U V g ' . / c 0 & g ∈ U V ' O ' P K ' !$ $ V ' $ c ; g ∈ V { } O P K #
K #) g O P / # 0 # 1 μH . ' ($ 8 μH (K) = h ∈ G ' 1 ! K K ⊂ h(K) & ($ 8 $ ) h ' h c μH ! M $ 1 " $ G h ' & . # /4 = 1 0 $ c < 1 # g ' μH (g(K) ∩ K) < c ! $ G ' c < 1 $
μH N μH U μ $ K ⊂ h(K) . h ∈ G ' g ' μ(g(K) ∩ K) < c !$ /. '0 ' / c < 1 0 ( ! K U ⊂ H g ∈ U & g(K) ⊃ K ⇐⇒ g = " ! - n K ? $ ' & $ U g ∈ U g(K) ⊂ K ⇐⇒ g = ! g ∈ U K ⊂ g−1 (K) ⇐⇒ g = U = U −1 = {g−1 : g ∈ U } U h ∈ U K
⊂ h(K) ⇐⇒ h = $ " 4 < 6 $ ϕI1 (K), . , ϕIk (K) < # M $$' # k = n + 1 # M ? . # K ? $ 1 d = min1≤i≤k (ϕIi (K), K ϕIi (K)) d > 0 1 U g ∈ U ' K # d 1 g ∈ U g(K) ⊂ K ! d U L)' g(ϕIi (K)) ⊂ ϕIi (K) 1 ≤ i ≤ k " ' M )' ' ) # $ M $$ # & M $ ' $ ' . 3 $B LM# LM# g " g LM# ? k = n + 1 LM# ? $ ## K ? $
g = ( ! K = ϕ1 (K) . ϕr (K) c<1 U ⊂ H μ g ∈ U { } =⇒ μ(K ∩ g(K)) < c " ! N 5 + 6 U ) g ' g(K) K g ∈ U x ∈ K (x, g(x)) ≤ 1 N U # H # 1 V ⊂ U i = 1, . , r −1 ϕi ◦ V ◦ ϕi ⊂ U i = j g ∈ V g(ϕi (K)) ∩ ϕj (K) = ∅ & $ # )' . g # ' & $ c < 1 g ∈ U V =⇒ μ(g(K) ∩ K) < c % . U V
' gn μ(K ∩ gn (K)) 1 & U V # gn gni gni h(K) ⊃ K U V ) h ' h ∈ U V & 4 = 1 7$ g ∈ U { } μ(K ∩ g(K)) < c $ c ( g ∈ U V $ 1 g ∈ V { } - A g LM# 3 $ x ∈ K A ϕIx (K) x ' ϕ−1 Ix ◦ g ◦ ϕIx ∈ U V 1 (i1 , i2 , . ) M {x} = ∞ (ϕi1 ◦ ϕi2 ◦ . ◦ ϕin )(K) n=1 1 In = (i1 , . , in ) " g ∈ V ϕ−1 i1 ◦ g ◦ ϕi1 ∈ U V −1 n ϕIn ◦ g ◦ ϕIn ∈ V V L)' L)' . ( −1 −1 ϕ−1 In+1 ◦ g ◦ ϕIn+1 = ϕin+1 ◦ ϕIn ◦ g ◦ ϕIn ◦ ϕin+1 ∈ U. @ −1
3 $ n ϕ−1 In ◦ g ◦ ϕIn ∈ V 1 y = ϕIn (x) y ∈ K −1 −1 {x} = ∞ LM# g n=1 ϕIn (K) (ϕIn ◦g ◦ϕIn )(y) = ϕIn (g(x)) " x n (g(x), x) = t > 0. g(x), ϕIn (K) > 2 - ϕi αi 2 ϕI αI (0 < αI < 1) ! ϕ−1 (ϕ−1 In (g(x)), K > t/αIn In ◦ g ◦ ϕIn )(y), K > t/αIn n 1 n ϕ−1 V In ◦ g ◦ ϕIn U " # n ϕ−1 In ◦ g ◦ ϕIn ∈ U V n Ix = In & $ g(ϕIx (K))∩K ⊂ ϕIx (K) $ $ k < n ϕ−1 Ik ◦g ◦ϕIk ∈ V (ϕ−1 Ik ◦ g ◦ ϕIk ◦ ϕj )(K) ∩ ϕl (K) = ∅ j = l F $ j ik+1 −1 (ϕIk ◦ g ◦ ϕIk+1 )(K) ∩ ϕl (K) = ∅
l = ik+1 ## (g ◦ ϕIk+1 )(K) ∩ (ϕIk ◦ ϕl )(K) = ∅ /l = ik+1 0 #) (g ◦ ϕIn )(K) ∩ (ϕIk ◦ ϕl )(K) = ∅ (l = ik+1 ). ! k = 0 1 . n − 1 ## g(ϕIn (K)) ∩ K ⊂ ϕIn (K) & {ϕIx (K) : x ∈ K A} . K A " x = y ϕIx (K) ∩ ϕIy (K) = ∅ ϕIx (K) ⊂ ϕIy (K) ϕIx (K) ⊃ ϕIy (K) . ' ' 2 $ . K A 1 K A ⊂ ∞ i=1 ϕJi (K) . ! g(ϕJi (K) ∩ K) ⊂ ϕJi (K) A ' K ? 4 < 1 # μ(g(K) ∩ K) ≤ μ(g(A) ∩ K) + μ(g(K A) ∩ K) = μ(g(K A) ∩ K) ≤ ∞ ∞ ∞ ϕJi (K) ∩ K = μ g(ϕJi (K)) ∩ K = μ g(ϕJi (K)) ∩ ϕJi (K) = ≤μ g i=1 i=1 i=1 ∞ ∞ −1 μ ϕJi ((ϕJi ◦ g ◦ ϕJi )(K) ∩ K) = pJi μ (ϕ−1 = Ji ◦ g ◦ ϕJi )(K) ∩ K
. i=1 i=1 " ϕ−1 c Ji ◦ g ◦ ϕJi ∈ U V $ ' . # μ(g(K) ∩ K) < pJi · c = μ(ϕJi (K)) · c = μ ( ϕJi (K)) · c = c ! $ ( ! K = ϕ1 (K) . ϕr (K) # g ∈ G G ⊂ H g & g (K) ∩ K = g(K) ∩ K " ! & K ? Rd 1 δ = min1≤i<j≤r (ϕi (K), ϕj (K)) D = (K) K ( g ' D/δ g (K) )' g (ϕj (K)) +D D ϕi g (K) ∩ K ⊂ g (ϕi (K)) ! g ' ) g ◦ ϕi ' Rd ' x Ax + b
' A M ' b # F b /Rd 0 g(K) ∩ K = ∅ g(x) = Ax + b . D/δ ' N Rd ' ' ' / M 0 # G # ( ! K = ϕ1 (K) . ϕr (K) * + ! μH μH (K) = 1 g g(K) ⊃ K c < 1 ,-& g μH (g(K) ∩ K) < c # K ⊂ g(K) ,- . ϕI (K) c g & μH g(ϕI (K)) ∩ ϕI (K) < c · μH ϕI (K) # ϕI (K) ⊂ g(ϕI (K)) " ! &
' ' $ 1 ε > 0 1 h ' K ⊂ h(K) - Kδ K δ $ " μH h(K) ) δ > 0 μH Kδ ∩ (h(K) K) < ε & $ 5 4 6 K μH U cH # 1 W ⊂ H W ∪ W −1 ⊂ U g ∈ W ∪ W −1 =⇒ μH (g(H)) ≤ x ∈ K (g(x), x) < δ (1 + ε)μH (H) H 3 1 g ∈ W h g = h W h h ! g ◦h−1 ∈ W { } h ◦ g−1 ∈ W −1 { } (h ◦ g−1 )(K) ⊂ Kδ ! μH K ∩ g(K) ≤ (1 + ε)μH (h ◦ g−1 )(K ∩ g(K)) = (1 + ε)μH (h ◦ g−1 )(K) ∩ h(K) = = (1 + ε)μH (h ◦ g−1 )(K) ∩ K + (1 + ε)μH (h ◦ g−1 )(K) ∩
(h(K) K) ≤ ≤ (1 + ε)cH + (1 + ε)μH Kδ ∩ (h(K) K) ≤ (1 + ε)cH + (1 + ε)ε. ! 1 ε ) $ h W h g = h ' g(K) ⊃ K ! $ g(K) ⊃ K ' & 4 = 1 ' ## . sup {μH (g(K) ∩ K) : g ∈ H, g(K) ⊃ K} = 1 ! sup {μH (g(K) ∩ K) : g ∈ G, g(K) ⊃ K} = 1 1 gn G gn (K) ⊃ K μH (gn (K) ∩ K) 1 gn h ! 4 = 1 h(K) ⊃ K gn = h ! n gn ∈ W h μH (K ∩gn (K)) ≤ (1+ε)cH +(1+ε)ε μH (gn (K) ∩ K) 1 ++ * ) ! # # " ! 1 U # K " x ∈ U Ex x ∈ Ex
⊂ U " U # Ex . & . + . K = ϕ1 (K) ϕr (K) μ g c < 1 μ(g(K) ∩ K) < c # K ⊂ g(K) " ! ! ) G ' 5 5 6 1 cH 5 9 6 # K μH 1 h ∈ G ' h(K) ⊃ K & h /5 9 6 # 0 G # & $ 4 : 1 h(K) ' μH 0 < ε ≤ 1 − cH H = K ⊂ h(K) ! ϕI μH K ∩ h(ϕI (K)) ≥ (1 − ε) μH h(ϕI (K)) . & 5 : h(ϕI (K))
K ' # K ⊃ h(ϕI (K)) " h(ϕI (K)) h(K) K h(ϕI (K)) # 5 J 1 h(ϕI (K)) K $' ! ϕJ (K) ϕJ (K) ⊂ h(ϕI (K)) ⊂ K ⊂ h(K) " ϕJ (K) K h(ϕI (K)) h(K) ! (ϕJ (K), h(K) ϕJ (K)) > 0 h g g ◦ h−1 h(K) ϕJ (K) ∩ ϕJ (K) = ∅. ! g g(K) ∩ ϕJ (K) = g ◦ h−1 h(K) ∩ ϕJ (K) = = g ◦ h−1 ϕJ (K) ∩ ϕJ (K) g ◦ h−1 h(K) ϕJ (K) ∩ ϕJ (K) = = g ◦ h−1 ϕJ (K) ∩ ϕJ (K). & 5 4 6 # $ h g g = h μ g ◦ h−1 ϕJ (K) ∩ ϕJ (K) < c · μ ϕJ (K) = c · pJ . μ g(K) ∩ ϕJ (K) = μ (g ◦ h−1 )(ϕJ (K)) ∩ ϕJ (K) < c · pJ
μ g(K) ∩ K = μ g(K) ∩ ϕJ (K) + μ g(K) ∩ (K ϕJ (K)) < c · pJ + 1 − pJ = 1 − (1 − c)pJ . +4 " ) h ' ' c < 1 G h # g μ g(K) ∩ K < c ! /. 4 = 1 0 c < 1 g ∈ G μ g(K) ∩ K < c g(K) ⊃ K / 5 9 6 0 G ' ! K ' ϕI (K) $ & ϕI (K) $ c ' K μ g(K) ∩ ϕI (K) < c · μ ϕI (K) ϕI (K) ⊂ g(K) ,- K = ϕ1 (K) . ϕr (K) μ K & ! & g μ g(K) ∩ K > 0 g(K) ∩ K #
K & μ K (g(K) ∩ K) = μ g(K) ∩ K % % g " ! ( g(K) ∩ K # # # 2 1 c 5 = K ' 1 g ' μ g(K) ∩ K > 0 & $ 4 J 1 ε = 1 − c 1 ' $ ai (K) μ g(K) ∩ ai (K) = μ (g(K) ∩ K) ∩ ai (K) > c · μ ai (K) g(K) ∩ K i ai (K) $ 2 & 5 = ai (K) ⊂ g(K) " ai (K) K g(K) ∩ K i ai (K) ⊂ K (g(K) ∩ K) ! μ g(K) ∩ K = μ g(K) ∩ K ∩ ai (K) + μ (g(K) ∩ K) i ai (K) = ai (K) ≤ μ K (g(K) ∩ K) , =μ i ! " #
& . # 2 ' ) K = ϕ1 (K) . ϕr (K) / g # g(K) ⊂ K g(K) 0 " ! & g ' $' & . K ⊂ g−1 (K) & $' $ ' 5 9 6 # +5 & h ∈ H ' LM h(K) ∩ K = ∅ # H / 5 5 6 0 " # . ) K = ϕ1 (K) . ϕr (K) # λ(K) ⊂ K ! k ≥ 1
I, J λ 1 k λ ◦ ϕI = ϕJ " ! " k ≥ 1 ϕI (K) k λk (K) ! I (ϕ−1 I ◦ λ )(K) ⊂ K )' % ' ' ) 9 + 1 −1 −1 k k k −k λk −k ◦ ϕ = ϕ k < k I I ϕI ◦ λ = ϕI ◦ λ ϕI ◦ ϕ−1 I I I =λ / - & λ(K) ⊂ K λ ' μ(λ) μ λ(K) ( ' # )' $ ◦ g1 g2 g1 ◦ g2 gk k g # )' & μ ' 2 ) . ' ' ϕI ' μ ◦ λ(K) ⊂ K $ λ
' μ ◦ λ = μ(λ) · μ ϕI = μ(ϕI ) · μ & ( ! K = ϕ1 (K) . ϕr (K) μ $ * ! # + # λ(K) ⊂ K μ ◦λ = μ λ(K) ·μ % μ λ(H) = μ λ(K) · μ H λ H ⊂ K " ϕI (K) μ g(ϕI (K)) ( g # g(ϕI (K)) ⊂ K μ ϕI (K) = " ! & λ ' # # & ' $ & 9 + 1 ) λ ' λ(K) ⊂ K λ(K) )' - λ0 , λ1 , . , λt
λ0 & $ ' F' λ ' λ(K) ⊂ K ϕI ◦ λi I i /$ ϕI (K) λ(K) ϕ−1 I ◦ λ ' K # )' +9 λi 0 & # μ ◦ ϕJ = pJ · μ = μ(ϕJ (K)) · μ J $ M H ⊂ K 3 μ λ(H) = μ (ϕI ◦ λi )(H) = μ ϕI (K) · μ λi (H) = μ ϕI (K) · μ λi (K) · μ H = = μ (ϕI ◦ λi )(K) · μ H = μ λ(K) · μ H , $ " 0 ≤ i ≤ t 9 4 1 Ii Ji $ M ki # λki i ◦ ϕIi = ϕJi 1 bi = ϕIi ci = ϕJi λki i bi = ci
1 μ∗ (λi ) = ki μ(ci ) . μ(bi ) 7$ μ∗ (λi ) = μ(λi ) " 0 ≤ i ≤ t I $ M 0 ≤ j ≤ t J $ M λi ◦ϕI = ϕJ ◦λj / ϕJ (K) (λi ◦ϕI )(K)0 ( ' # ))'; g1 g2 ' g1 ≈ g2 g1 ◦ g2−1 ' H g1 (H) g2 (H) ' g1 g2 . ! ' . # g1 ◦ g2 ≈ g3 ⇐⇒ g2 ◦ g1 ≈ g3 k $ λi ϕI = ϕJ λj λki i bi = ci λj j bj = cj ' kk kk k k k k kk k k k λi i j ϕI i j bi j bkj i ≈ ϕJi j bi j λj i j bkj i ≈ ϕJi j bi j ckj i , k k k k k k k k k ≈ϕJi j λj i j ≈cj i k kk k k k k kk λi i j ϕI i j bi j bkj i ≈ λi i j bi
j ϕI i j bkj i ≈ ci j ϕI i j bkj i . k ≈ci j & kk k k k k ϕJi j bi j ckj i ≈ ci j ϕI i j bkj i . & bi bj ci cj . ' ' # )' ' k k k k k k ϕJi j bi j ckj i (K) ci j ϕI i j bkj i (K) ' . & μ(ϕJ )ki kj μ(bi )kj μ(cj )ki = μ(ci )kj μ(ϕI )ki kj μ(bj )ki , μ∗ L)' # μ(ϕJ )ki kj μ∗ (λj )ki kj = μ∗ (λi )ki kj μ(ϕI )ki kj , μ∗ (λj )μ(ϕJ ) = μ∗ (λi )μ(ϕI ). μ∗ (λi )μ ϕI μ λj . μ(λi ϕI ) = μ(ϕJ λj ) = μ(ϕJ )μ(λj ) = μ∗ λj +: ! ) ; i I j μ(λi ϕI ) = μ∗ (λi ) μ(λj ) μ(ϕI ). μ∗ (λj ) ! μ∗ (λj ) = 0 L)' . 1 m M μ(λm )
˙ μ(λi ) ≤ ∗ μ∗ (λm ) μ (λi ) 0 ≤ i ≤ t M Q # $ $ $ ! ϕI μ(λm ϕI ) = μ∗ (λm ) μ(λj ) ˙ μ∗ (λm ) μ(λm ) μ(ϕI ) = μ(λm )μ(ϕI ) μ(ϕI ) ≥ ∗ μ (λj ) μ∗ (λm ) 0 ≤ j ≤ t 1 {ϕIi (K)} ' ' #)' K ϕI (K) # ! μ λm (K) = μ λm ϕIi (K) ˙ μ λm ϕIi ≥ ˙ ≥ μ(λm )μ(ϕIi ) = μ(λm ), =μ λm (ϕIi (K)) = μ(λm ϕI ) = μ(λm )μ(ϕI ) I $ M 1 H ⊂ K 3 & μ L)' . aij (K) H ⊂ j i aij (K) μ(H) = inf j μ i aij (K) = μ j i aij (K) ! μ(λm (H)) ≤ μ λm j i j = μ(λm ) inf j j i R H c = K H =μ λm (aij (K)) = inf ≤ inf μ j aij (K)
λm (aij (K)) ≤ i μ(λm aij ) = inf j i μ(aij ) = μ(λm )μ i i j i μ(λm )μ(aij ) = aij (K) = μ(λm )μ(H). μ λm (H c ) ≤ μ(λm )μ(H c ) & μ λm (H)) + μ λm (H c ) ≤ μ(λm )μ(H) + μ(λm )μ(H c ) ' H μ λm (H) = μ(λm )μ(H) μ ◦ λm = μ(λm ) · μ ! # $ H ⊂ K 3 n−1 n−1 μ λnm (H) = μ λm (λm (H)) = μ(λm )μ λm (H) , $)' μ λnm (H) = μ(λm )n μ(H) #) μ(λnm ) = μ(λm )n ! μ(λkmm bm ) = μ(λm )km μ(bm ) & μ∗ (λm ) L)' μ∗ (λm )km μ(bm ) cm = λkmm bm μ(cm ) μ(λm )km μ(bm ) = μ(λkmm bm ) = μ(cm ) = μ∗ (λm )km μ(bm ). +J = i) " μ(bm ) > 0 # $ μ(λm ) = μ∗ (λm ) " m i M μμ(λ ∗ (λ ) i 0 ≤ i ≤ t μ∗
(λi ) ≤˙ μ(λi ) m) & m M μμ(λ ∗ (λ ) ≥ m μ(λi ) 0 ≤ i ≤ t ) # . $ ∗ μ (λi ) # $ 0 ≤ i ≤ t μ∗ (λi ) ≥ μ(λi ) ! 0 ≤ i ≤ t μ∗ (λi ) = μ(λi ) ! m ' 0 ≤ i ≤ t $ i μ ◦ λi = μ(λi ) · μ & λ(K) ⊂ K λ ' μ ◦ λ = μ(λ) · μ μ ◦ λn = μ(λ)n · μ " # . ϕL (K) g ' K # g(ϕL (K)) ⊂ K & 9 4 1 I J $ M k # (g ◦ ϕL )k ◦ ϕI = ϕJ ! # μ(ϕJ ) = μ (g ◦ ϕL )k ◦ ϕI = μ(g ◦
ϕL )k μ(ϕI ), ϕJ = (g ◦ ϕL )k ◦ ϕI ≈ (ϕL )k ϕI μ(ϕJ ) = μ((ϕL )k ϕI ) = μ(ϕL )k μ(ϕI ), μ(g ◦ ϕL )k μ(ϕI ) = μ(ϕL )k μ(ϕI ), μ(g ◦ ϕL ) = μ(ϕL ), $ . 0 ' 1 2 K = ϕ1 (K) . ϕr (K) # # μ $ ! μ # & K & " ! & $ g ' H ⊂ K 3 g(H) ⊂ K μ(H) = μ(g(H)) 1 c 5 = c < 1 ! H ⊂ K # 2 & $ 4 : 1 ε = 1−c H # $ a(K) μ H ∩ a(K) > c · μ a(K) " H ⊂ g−1 (K) μ g−1 (K) ∩ a(K) > c · μ a(K)
5 = a(K) ⊂ g−1 (K) g(a(K)) ⊂ K 1 λ = g ◦ a & 9 5 6 # μ(λ) = μ(a) / μ(λ) μ λ(K) 0 H0 = a−1 (a(K) ∩ H) μ(λ(H0 )) = μ(λ)μ(H0 ) 0 < c · μ a(K) < μ a(K) ∩ H = μ a(H0 ) = μ(a)μ(H0 ) = μ(λ)μ(H0 ) = = μ λ(H0 ) = μ g(a(H0 )) = μ g(a(K) ∩ H) ≤ μ g(H) , += g(H) # 2 F # 2 ' # # 2 ## $ 2 ' # $ 2 1 H ⊂ K 3 g # ' g(H) ⊂ K & $ 4 J 1 0 < ε < 1 − c ! ai (K) μ H ∩ ai (K) > (1 − ε) · μ ai (K) μ H ai (K) = 0. i ! H ⊂ g−1 (K) μ g−1 (K) ∩ ai (K) > (1 − ε) · μ ai (K) & 5 = # g−1
(K) ⊃ ai (K) g(ai (K)) ⊂ K & 9 5 6 μ g(ai (K)) = μ ai (K) . $ 2 ' # $ 2 μ g(H) = μ g H ∩ ai (K) +μ g H i = ai (K) =μ g H∩ i ai (K) = i μ g(H ∩ ai (K)) ≤ μ g(ai (K)) = μ ai (K) ≤ i i 1 1 · ·μ H ∩ μ H ∩ ai (K) = ≤ 1−ε 1−ε i i ai (K) = i 1 · μ(H). 1−ε ! 0 < ε < 1 − c μ g(H) ≤ μ H R H g(H) g g−1 μ H ≤ μ g(H) μ H = μ g(H) μ ' 1 " 4 2 ' - αi ϕi ' F # ' K ' # μ
# ' αi1 αi2 . αin = αj1 αj2 αjm pi1 pi2 . pin = pj1 pj2 pjm μ pi & αi pi $ #$ − log pi ' − log αi (i = 1, . , r) # ' . ) Q . #) r . Q ' & ' − log pi ' # r ' Rr # ' / ' 0 i pi = 1 . # ( 1 ' # 1 # ($ 8 $ $ /%
' pi = αsi $ $ 0 & . ' & ' +< . K μ # &# % # 2 {ϕ1 , ϕ2 , . , ϕr } # &# ! μ " ! 1 {ψ1 , . , ψs } & 9 4 1 k # ϕI ϕJ ψik ◦ ϕI = ϕJ 9 5 6 # # 0 < μ(ϕJ ) = μ(ψik ◦ ϕI ) = μ(ψi )k μ(ϕI ), μ(ψi ) > 0 1 ≤ i ≤ s & 9 5 6 # μ ◦ ψi = μ(ψi ) · μ μ ψi (K) = 1 μ {ψ1 , . ,
ψs } # & )' 2 * ,- K = ϕ1 (K) . ϕr (K) μ # ! 0 # μ # & & # " ! & 9 9 $ ' 1 ϕI (K) ϕJ (K) ' ! ϕ2I (K) ϕ2J (K) ' ϕ2I ϕ2J ' ϕ2I ◦ ϕ−2 J . 2 2 2 μ(ϕI (K)) = μ(ϕJ (K)) & μ(ϕI (K)) = μ(ϕI (K))2 μ(ϕ2J (K)) = μ(ϕJ (K))2 μ(ϕI (K)) = μ(ϕJ (K)) ! $ $ % ! $$ ' / $ '0
) . 1 ϕ1 (x) = x3 ϕ2 (x) = x3 + 23 K # K = ϕ1 (K) ∪ ϕ2 (K) 7 1 ψ1 (x) = − x3 + 13 ! K = ψ1 (K) ϕ2 (K) . ) ' ' ! . O P; {ϕ1 (K), ϕ2 (K)} {ψ1 (K), ϕ2 (K)} ' )' {a1 (K), . , ar (K)} {b1 (K), . , bs (K)} j = 1, s 1 ≤ i ≤ r bj (K) ⊂ ai (K) ! . $ ' ) $ 2 7 . +@ & $ . $
' ) . K ' ; ( λ1 λ2 ' λ1 (K) ⊂ K λ2 (K) ⊂ K λ1 (K) ∩ λ2 (K) = ∅ λ1 (K) ⊂ λ2 (K) λ2 (K) ⊂ λ1 (K) & 5 2 $ ' ' ' & $)' $ " . K ) . 1 a, b, c # ' a + b + a + c + a + b + a = 1 b = a · c ! 2 0 < a < 1/4 b c . 1 ϕ1 [0, 1] [0, a] # ' ' F ϕ2 [0, 1] [a+b, a+b+a] ϕ3
[1−a−b−a, 1−a−b] ϕ4 # [1−a, 1] ' F ϕ1 (x) = a·x ϕ2 (x) = a·x+a+b ϕ3 (x) = a · x + 1 − a − b − a ϕ4 (x) = a · x + 1 − a D + ϕ1 ([0, 1]) ϕ2 ([0, 1]) ) ψ1 ([0, 1]) ϕ3 ([0, 1]) ϕ4 ([0, 1]) ψ2 ([0, 1]) 1 K # K = ϕ1 (K)ϕ2 (K)ϕ3 (K)ϕ4 (K) ! K )' a 2 b c $ F K ⊂ [0, 1] K $ 12 & K . # ϕi ([0, 1]) [0, 1] ' ' /a = 0, 15 c = 0,4 1,3 0 & & ϕi (ϕj ([0, 1])) (1 ≤ i, j ≤ 4) . & K #'$ ' 1 ψ1 (x) = a · x + a2 + a · b + a2 + a · c ψ2 (x) = a · x + 1 − a − b
− a2 − a · b − a2 & [0, 1] ϕ21 ϕ1 ◦ ϕ2 ψ1 ϕ2 ◦ ϕ3 ϕ2 ◦ ϕ4 ϕ3 ◦ ϕ1 ϕ3 ◦ ϕ2 ψ2 ϕ4 ◦ ϕ3 ϕ24 ' # . & $ ψ1 (K) ⊂ K ψ2 (K) ⊂ K {ϕ21 , ϕ1 ◦ ϕ2 , ψ1 , ϕ2 ◦ ϕ3 , ϕ2 ◦ ϕ4 , ϕ3 ◦ ϕ1 , ϕ3 ◦ ϕ2 , ψ2 , ϕ4 ◦ ϕ3 , ϕ24 } K ' . / 0 ! $ ψ1 ◦ ϕ1 = ϕ1 ◦ ϕ3 ψ1 ◦ ϕ2 = ϕ1 ◦ ϕ4 ψ1 ◦ ϕ3 = ϕ2 ◦ ϕ1 ψ1 ◦ ϕ4 = ϕ2 ◦ ϕ2 4D ψ2 ◦ ϕ1 = ϕ3 ◦ ϕ3 ψ2 ◦ ϕ2 = ϕ3 ◦ ϕ4 ψ2 ◦ ϕ3 = ϕ4 ◦ ϕ1 ψ2 ◦ ϕ4 = ϕ4 ◦ ϕ2 ! 2 b = a · c " $ $ ) { 1 (K), . , r (K)} K ' K {ϕ1 , ϕ2 , ϕ3 , ϕ4 } . L .
' ! K 2 . /F' 0 % . { 1 (K), , r (K)} " ϕ1 (K) ∩ ψ1 (K) = ∅ i ϕ1 (K) ∪ ψ1 (K) ⊂ i (K) R i ϕ2 (K) ∪ ψ1 (K) ⊂ i (K) ( ' j M ϕ3 (K) ∪ ψ2 (K) ∪ ϕ4 (K) ⊂ j (K) ! K = i (K) j (K) i j ' ) [0, 1] [0, a + b + a] [1 − a − b − a, 1] # b = (a + b + a) · c . b = a · c & λ1 λ2 ' # $ ϕ1 ψ1 ' . & *+, - ! ($) N) : =+5S=9= *4, T$ E 3 2 ' ( . 3 # ( 4 3 /+@<+0 . $ ) 0 3
/+@@<0 +D 9DJ:S9D<= . 7 *5, 7 C$ & " -7 # T )) ) . $ 7 B ($ 8 ' ( 5+6 /4DD+0 + 9:S@J *9, N Q -$ 1 % ) . ) 7 B % N$ # # 7 80 /4DD40 @ +<D=S+<+= *:, UM 1 I V % ) . ( 0 5! ! 6 # /+@@@0 4 4+9S4+@ . 7 ) '0 *J, G H I B ) $ L . $ ( /+@=+0 +J+S+J4 *=, ! " 3 B ) $ σ L . $ '# ( *<, 3 . E I R . UB) +@<9 *@, 7$ ")"$ ($ #* /+@<90 +S@ 8 .
#A )# 4+ *+D, I C $ E 1 ($ 8 M .? ) /+@@40 :55S:J< *++, T$ E #) $ . ? )# ! /+@@90 5 95=S9:D . ) *+4, T$ E .? )# ")"$ 3 L ($ $ ! * /+@@90 :+5S:4J *+5, 1) ) "' / %# !1! +@@: 44 8
# K K = ϕ1 (K) ∪ . ∪ ϕr (K) - $ $' ! K ' . / $ ' 0 ϕ1 , ϕ2 , , ϕr ' K = ϕ1 (K) . ϕr (K) K ' # $ $' & . K ' ' {ϕ1 , ϕ2 , . , ϕr } K = ϕ1 (K) ϕr (K) & K . ' (ϕi1 ◦ ◦ ϕin )(K) ! . ' 1 r p1 , . pr ∈ (0, 1) μ ' 2 3 i=1 pi = 1 % K μ◦ϕi = pi ·μ (1 ≤ i ≤ r) H ⊂ K 3 μ(ϕi (H)) = pi · μ(H) (1 ≤ i ≤ r) # $ *+, 4 5 6
& . $ ' / 7 0 $ ' 1 $ ($ 8 # ' $ / # $ *4, 5, 9, :,0 . # 2 /0 & $ / $ 0 7 # # 2 7 $ /7 0 *9, % ' $; K ' $ ' μ g ' K ∩ g(K) K μ K ∩ g(K) = μ K (K ∩ g(K)) 5 < !
. / 0 ; " $ ' ' K ' μ c < 1 g ' μ K ∩ g(K) < c K ⊂ g(K) 5 = >? K = ξ1 (K) ∪ . ∪ ξr (K) # $ ξi ? / ? . )' # )' 0 . ? #) ? 3 . *<, ")"$ @, #A " B ($ 8 ' C $ 1 *+D, $ 2 ($ 8 $ E *+4, $ . + ($ 8 *++, # !
$ ' - . ! " #A ? ' # 2 /! #$ #A $ 0 F ' # 2 G H I *J, $ # $ & 1 $ $ *=, 1 K ⊂ Rd 3 μ K 3 & μ ' $ H ⊂ K 3 g ' g(H) ⊂ K μ(H) = μ(g(H)) ( ' ' $ /4 9 6 0;
μ∗ ' 3 Rd μ∗ |K = μ ! / ' ?0 ' K # 2 K . ' & . ? $ $ ! $ ' ' ' )' $ ' & . $ " $ ' ' K ⊂ Rd ' μH ' 3 Rd μH (K) = 1 . ' ($ 8 / # $ *+5, +D= 0 & ($ 8 ' ' '
μH K ' $ ' ' ' ! ($ 8 $$ 5 = ' )' F μ ' ' 2 " $ $ . ' /9 9 0 ! μ pi ' ' ' ϕi ' ' 2 ' μ ' & ' . . ' 9 : 4 !
$ # . L)' ! $ ' ' ' . # $ {ϕ1 , ϕ2 , . , ϕr } {ϕ1 ◦ ϕ1 , ϕ1 ◦ ϕ2 , , ϕ1 ◦ ϕr , ϕ2 , ϕ3 , , ϕr } & K {ϕ1 , . , ϕr } ' (ϕi1 ◦ ◦ ϕin )(K) (ϕi1 ◦. ◦ϕin )(K) )' F $ M ' ; M $ M I = (i1 , i2 , . , in ) # $ ϕI = ϕi1 ◦ ◦ ϕin pI = pi1 pi2 pin & I = ∅ $ M ϕ∅ = p∅ = 1 N K ' ϕI (K) ' h ' h(K) ' h(ϕI (K)) %& !' K K
" ! # $ μ ◦ ϕi = pi · μ μ(K) = 1 # μ p1 , . , pr ∈ (0, 1) {ϕ1 , ϕ2 , . , ϕr } # r i=1 pi = 1 (i = 1, . , r) 3 . $ σ # 3 ' 1$ ' ) 2 . O P . ) & $ F = {ϕ1 , ϕ2 , . , ϕr } L F1 = {ψ1 , ψ2 , . , ψs } 1 ≤ i ≤ r I $ M ϕi = ψI ! L . F1 ≤ F F L F1 ! . . ' L ! . L F ' / $ '0 . /
0 ) . / : 0 & . ) ' ' ' F . L %& !' K ⊂ Rd ' % " μ μ # &# g # g(H) ⊂ K K " & H ⊂ K " μ(H) = μ(g(H)) ! K = Rd L)' ' ) & 4 9 6 $ L)'; K ' Rd ' 5 - αi ϕi ' F # ' K ' # F ϕI (K) ϕJ (K) ' αI = αJ & 9 9
)' μ # ' αi1 αi2 . αin = αj1 αj2 αjm pi1 pi2 . pin = pj1 pj2 pjm μ pi " $ ' ' ' . ' ($ 8 # *+5, $ + / # ($ 8 ' 0 #$ # ' & ($ 8 . pi = αsi s # ' αsi = 1 ## ' ' & Rd $ ' ' ' # ' " ' x αAx + b α ≥ 0 ' A n × n M ' b ∈ Rd # ! 2 ' Rd +d
; d2 αA 2 M ' d # b ' F Rd +d 2 $ ' Rd +d 2 H ' $ " ' Rd +d H $ # # # & ' # )' $ ' # ) # # )' ! ) # # $ ) # L # ' /# ) # 0 (h, g) h ◦ g g g−1 . # ! 2 ' gn ∈ H ' # h ∈ H ' /gn h0 K ⊂ Rd # ε > 0 N n ≥ N x ∈ K gn (x), h(x) < ε %
dist # ( ! K = ϕ1 (K) . ϕr (K) ! 0 ≤ p1 , . , pr ≤ 1 " K & pi = 1 H ⊂ K " μ # $ 1 ≤ i ≤ r & μ(ϕi (H)) = pi · μ(H) # μ ◦ ϕi = pi · μ " ! F ϕI (K) ; μk (H) = inf pI ! L $ μk ∞ pI i : H ⊂ i=1 ∞ i=1 ϕIi (K) . & K $ $ ' $ ' # 3 $ 3 σ # μ 9 & $ μ(K) = 1 F μ(K) ≤ 1 F K .
' K ⊂ ∞ # i=1 ϕIi (K) & ϕIi (K) ) K . K F $ . ! M K = ni=1 ϕIi (K) ϕIi (K) ∩ ϕIj (K) = ∅ i = j ! . # $ M ' $)' # $ ni=1 pIi = 1 μ(K) = 1 & μk L)' # H ⊂ K 3 μ(ϕi (H)) = pi · μ(H) 1 ≤ i ≤ r 1 μ ' 2 3 μ ◦ ϕi = pi · μ (1 ≤ i ≤ r) & μ μ μ (ϕI ) = μ(ϕI ) = pI K ' $ $' μ μ μk L)' # μ(H) = inf {μ(G) : H ⊂ G } , μ (H) ≤ μ (G) = μ(G)
(H ⊂ G ). ! μ (H) ≤ μ(H) $ μ (K H) ≤ μ(K H) #$ H ⊂ K 3 μ (H) = μ(H) ( ! ∅ = K ⊂ Rd " μ∗ # &# " K & μ # &# " Rd & μ∗ |K = μ " ! ( H ⊂ Rd 3 . K ' ' # μ∗ ( . $ H = i Hi ' 3 #)' Hi ⊂ gi (K) gi ' 1 μ∗ (H) = μ(gi−1 (Hi )). i & $ ' L)' . #)' ' ' 1 H = j Hj Hj ⊂ gj (K) #)' H = i,j Hi ∩ Hj #)'; i μ(gi−1 (Hi )) = i μ(gi−1 (Hi ∩ Hj )) = j
= i j μ(gj−1 (Hi ∩ Hj )) = μ(gj−1 (Hj )). j F μ∗ . #)' ' ' 3 $ μ∗ μ∗ (∅) = 0 & σ ' $ Ai 3 ( . K ' ' # $' $ $' # Ai = j Hij Hij ⊂ gij (K) ! i Ai = i,j Hij #)'' ' μ∗ i Ai = i μ∗ (Ai ) : & # μ∗ |K = μ μ∗ ' " $ # ) K = ϕ1 (K) . ϕr (K) μ ε > 0& a(K) " ! " H . H ⊂ K # $ "
μ H ∩ a(K) > (1 − ε)μ a(K) ( # H (1 + ε) μ(H) > ' i ϕIi (K) μ(ϕIi (K)). i ( i (1 + ε) μ(H ∩ ϕIi (K)) > μ(ϕIi (K)), ϕIi . a ! (1+ε) μ H ∩ϕIi (K) ≤ μ(ϕIi (K)) (1 + ε) μ(H) = (1 + ε) μ H ∩ ϕIi (K) ≤ μ ϕi (K) , i . * ) K = ϕ1 (K) . ϕr (K) H i ai (K) μ H ⊂ K " ε > 0& μ H ∩ai (K) > (1−ε)μ ai (K) ( ai (K) $ " ! " ' ' $ ai (K) & n # /n = 0, 1, 2, . 0 $ / 20 an (K) an
(K) ∩ ai (K) = ∅ (0 ≤ i < n) μ H ∩ an (K) > (1 − ε)μ an (K) ( ) $ $ & $ H i ai (K) $ 2 . # 2 ! 4 : 1 a(K) μ (H i ai (K)) ∩ a(K) > # & (1 − ε)μ a(K) a(K) μ H ∩ a(K) > (1 − ε)μ a(K) ( # $ a(K) a(K) ) # ai (K) ( # a(K) $) $ ai (K) Q ai (K) ⊃ a(K) ai (K) ⊂ a(K) + ) K ⊂ Rd μ=K μ # $ " Rd & gn # gn h K) 1 !
μ(h(K) ∩ K) = 1 K ⊂ h(K) J μ(gn (K) ∩ " ! . μ(h(K) ∩ K) = q < 1 - h(K)ε h(K) ε $ " ∞ μ h(K)1/n ∩ n=1 (h(K)1/n ∩ K) = h(K) ∩ K μ K μ h(K) ∩ K = q 1 ε > 0 μ h(K)ε ∩ K ≤ 1+q 2 < 1 ! 1+q n gn (K) ⊂ h(K)ε μ gn (K) ∩ K ≤ 2 μ(gn (K) ∩ K) 1 μ(h(K) ∩ K) = 1 " μ ' K K # 2 & K h(K) $ 2 K ⊂ h(K) ) K = ϕ1 (K) . ϕr (K) μ K & ! A ) μ(A ∩ K) = 0 # A⊃K " ! " J $ M ϕJ (K) ? ## ' K ϕJ ' &
$' ϕJ (K) ⊂ K ' ! A ? A ⊃ K # A ⊃ ϕJ (K) F $ M ? . # K x1 , x2 , , xk 1 U1 , U2 , . , Uk M # # k # ? . " 1 ≤ i ≤ k ϕIi (K) Ui 1 ε = min1≤i≤k μ(ϕIi (K)) > 0 F K ' A ? ϕIi (K) K $ J $ M A ? (ϕJ ◦ ϕIi )(K) 1 ≤ i ≤ k A ⊃ ϕJ (K) A ⊃ K ! μ(A ∩ ϕJ (K)) ≤ (1 − ε)μ(ϕJ (K)) A ⊃ K . A ⊃ K μ(A ∩ K) > 0 & 4 : 1 ϕJ (K) (1 − ε)μ(ϕJ (K)) < μ(A ∩ ϕJ (K)), . &
. #$ $ K ' K g(K) ' # /g ' 0 E $ K ⊂ g(K) & 4 < 6 g ' K A ? μ(g(K) ∩ K) = 0 K ⊂ g(A) ! ) ' . $ A LM ' ) A ' ' $ ' ? $ ' H " ## ' . K ? $ $ ! . ' . =
! . $; " $ ' ' K ' μ c < 1 g ' μ(g(K) ∩ K) < c K ⊂ g(K) & ) ' #' $ . 3 $ ' G # g ∈ H g ∈ G g (K) ∩ K = g(K) ∩ K 2 G ' $ V, U ⊂ H V ⊂ U ⊂ H U V g ' . / c 0 & g ∈ U V ' O ' P K ' !$ $ V ' $ c ; g ∈ V { } O P K #
K #) g O P / # 0 # 1 μH . ' ($ 8 μH (K) = h ∈ G ' 1 ! K K ⊂ h(K) & ($ 8 $ ) h ' h c μH ! M $ 1 " $ G h ' & . # /4 = 1 0 $ c < 1 # g ' μH (g(K) ∩ K) < c ! $ G ' c < 1 $
μH N μH U μ $ K ⊂ h(K) . h ∈ G ' g ' μ(g(K) ∩ K) < c !$ /. '0 ' / c < 1 0 ( ! K U ⊂ H g ∈ U & g(K) ⊃ K ⇐⇒ g = " ! - n K ? $ ' & $ U g ∈ U g(K) ⊂ K ⇐⇒ g = ! g ∈ U K ⊂ g−1 (K) ⇐⇒ g = U = U −1 = {g−1 : g ∈ U } U h ∈ U K
⊂ h(K) ⇐⇒ h = $ " 4 < 6 $ ϕI1 (K), . , ϕIk (K) < # M $$' # k = n + 1 # M ? . # K ? $ 1 d = min1≤i≤k (ϕIi (K), K ϕIi (K)) d > 0 1 U g ∈ U ' K # d 1 g ∈ U g(K) ⊂ K ! d U L)' g(ϕIi (K)) ⊂ ϕIi (K) 1 ≤ i ≤ k " ' M )' ' ) # $ M $$ # & M $ ' $ ' . 3 $B LM# LM# g " g LM# ? k = n + 1 LM# ? $ ## K ? $
g = ( ! K = ϕ1 (K) . ϕr (K) c<1 U ⊂ H μ g ∈ U { } =⇒ μ(K ∩ g(K)) < c " ! N 5 + 6 U ) g ' g(K) K g ∈ U x ∈ K (x, g(x)) ≤ 1 N U # H # 1 V ⊂ U i = 1, . , r −1 ϕi ◦ V ◦ ϕi ⊂ U i = j g ∈ V g(ϕi (K)) ∩ ϕj (K) = ∅ & $ # )' . g # ' & $ c < 1 g ∈ U V =⇒ μ(g(K) ∩ K) < c % . U V
' gn μ(K ∩ gn (K)) 1 & U V # gn gni gni h(K) ⊃ K U V ) h ' h ∈ U V & 4 = 1 7$ g ∈ U { } μ(K ∩ g(K)) < c $ c ( g ∈ U V $ 1 g ∈ V { } - A g LM# 3 $ x ∈ K A ϕIx (K) x ' ϕ−1 Ix ◦ g ◦ ϕIx ∈ U V 1 (i1 , i2 , . ) M {x} = ∞ (ϕi1 ◦ ϕi2 ◦ . ◦ ϕin )(K) n=1 1 In = (i1 , . , in ) " g ∈ V ϕ−1 i1 ◦ g ◦ ϕi1 ∈ U V −1 n ϕIn ◦ g ◦ ϕIn ∈ V V L)' L)' . ( −1 −1 ϕ−1 In+1 ◦ g ◦ ϕIn+1 = ϕin+1 ◦ ϕIn ◦ g ◦ ϕIn ◦ ϕin+1 ∈ U. @ −1
3 $ n ϕ−1 In ◦ g ◦ ϕIn ∈ V 1 y = ϕIn (x) y ∈ K −1 −1 {x} = ∞ LM# g n=1 ϕIn (K) (ϕIn ◦g ◦ϕIn )(y) = ϕIn (g(x)) " x n (g(x), x) = t > 0. g(x), ϕIn (K) > 2 - ϕi αi 2 ϕI αI (0 < αI < 1) ! ϕ−1 (ϕ−1 In (g(x)), K > t/αIn In ◦ g ◦ ϕIn )(y), K > t/αIn n 1 n ϕ−1 V In ◦ g ◦ ϕIn U " # n ϕ−1 In ◦ g ◦ ϕIn ∈ U V n Ix = In & $ g(ϕIx (K))∩K ⊂ ϕIx (K) $ $ k < n ϕ−1 Ik ◦g ◦ϕIk ∈ V (ϕ−1 Ik ◦ g ◦ ϕIk ◦ ϕj )(K) ∩ ϕl (K) = ∅ j = l F $ j ik+1 −1 (ϕIk ◦ g ◦ ϕIk+1 )(K) ∩ ϕl (K) = ∅
l = ik+1 ## (g ◦ ϕIk+1 )(K) ∩ (ϕIk ◦ ϕl )(K) = ∅ /l = ik+1 0 #) (g ◦ ϕIn )(K) ∩ (ϕIk ◦ ϕl )(K) = ∅ (l = ik+1 ). ! k = 0 1 . n − 1 ## g(ϕIn (K)) ∩ K ⊂ ϕIn (K) & {ϕIx (K) : x ∈ K A} . K A " x = y ϕIx (K) ∩ ϕIy (K) = ∅ ϕIx (K) ⊂ ϕIy (K) ϕIx (K) ⊃ ϕIy (K) . ' ' 2 $ . K A 1 K A ⊂ ∞ i=1 ϕJi (K) . ! g(ϕJi (K) ∩ K) ⊂ ϕJi (K) A ' K ? 4 < 1 # μ(g(K) ∩ K) ≤ μ(g(A) ∩ K) + μ(g(K A) ∩ K) = μ(g(K A) ∩ K) ≤ ∞ ∞ ∞ ϕJi (K) ∩ K = μ g(ϕJi (K)) ∩ K = μ g(ϕJi (K)) ∩ ϕJi (K) = ≤μ g i=1 i=1 i=1 ∞ ∞ −1 μ ϕJi ((ϕJi ◦ g ◦ ϕJi )(K) ∩ K) = pJi μ (ϕ−1 = Ji ◦ g ◦ ϕJi )(K) ∩ K
. i=1 i=1 " ϕ−1 c Ji ◦ g ◦ ϕJi ∈ U V $ ' . # μ(g(K) ∩ K) < pJi · c = μ(ϕJi (K)) · c = μ ( ϕJi (K)) · c = c ! $ ( ! K = ϕ1 (K) . ϕr (K) # g ∈ G G ⊂ H g & g (K) ∩ K = g(K) ∩ K " ! & K ? Rd 1 δ = min1≤i<j≤r (ϕi (K), ϕj (K)) D = (K) K ( g ' D/δ g (K) )' g (ϕj (K)) +D D ϕi g (K) ∩ K ⊂ g (ϕi (K)) ! g ' ) g ◦ ϕi ' Rd ' x Ax + b
' A M ' b # F b /Rd 0 g(K) ∩ K = ∅ g(x) = Ax + b . D/δ ' N Rd ' ' ' / M 0 # G # ( ! K = ϕ1 (K) . ϕr (K) * + ! μH μH (K) = 1 g g(K) ⊃ K c < 1 ,-& g μH (g(K) ∩ K) < c # K ⊂ g(K) ,- . ϕI (K) c g & μH g(ϕI (K)) ∩ ϕI (K) < c · μH ϕI (K) # ϕI (K) ⊂ g(ϕI (K)) " ! &
' ' $ 1 ε > 0 1 h ' K ⊂ h(K) - Kδ K δ $ " μH h(K) ) δ > 0 μH Kδ ∩ (h(K) K) < ε & $ 5 4 6 K μH U cH # 1 W ⊂ H W ∪ W −1 ⊂ U g ∈ W ∪ W −1 =⇒ μH (g(H)) ≤ x ∈ K (g(x), x) < δ (1 + ε)μH (H) H 3 1 g ∈ W h g = h W h h ! g ◦h−1 ∈ W { } h ◦ g−1 ∈ W −1 { } (h ◦ g−1 )(K) ⊂ Kδ ! μH K ∩ g(K) ≤ (1 + ε)μH (h ◦ g−1 )(K ∩ g(K)) = (1 + ε)μH (h ◦ g−1 )(K) ∩ h(K) = = (1 + ε)μH (h ◦ g−1 )(K) ∩ K + (1 + ε)μH (h ◦ g−1 )(K) ∩
(h(K) K) ≤ ≤ (1 + ε)cH + (1 + ε)μH Kδ ∩ (h(K) K) ≤ (1 + ε)cH + (1 + ε)ε. ! 1 ε ) $ h W h g = h ' g(K) ⊃ K ! $ g(K) ⊃ K ' & 4 = 1 ' ## . sup {μH (g(K) ∩ K) : g ∈ H, g(K) ⊃ K} = 1 ! sup {μH (g(K) ∩ K) : g ∈ G, g(K) ⊃ K} = 1 1 gn G gn (K) ⊃ K μH (gn (K) ∩ K) 1 gn h ! 4 = 1 h(K) ⊃ K gn = h ! n gn ∈ W h μH (K ∩gn (K)) ≤ (1+ε)cH +(1+ε)ε μH (gn (K) ∩ K) 1 ++ * ) ! # # " ! 1 U # K " x ∈ U Ex x ∈ Ex
⊂ U " U # Ex . & . + . K = ϕ1 (K) ϕr (K) μ g c < 1 μ(g(K) ∩ K) < c # K ⊂ g(K) " ! ! ) G ' 5 5 6 1 cH 5 9 6 # K μH 1 h ∈ G ' h(K) ⊃ K & h /5 9 6 # 0 G # & $ 4 : 1 h(K) ' μH 0 < ε ≤ 1 − cH H = K ⊂ h(K) ! ϕI μH K ∩ h(ϕI (K)) ≥ (1 − ε) μH h(ϕI (K)) . & 5 : h(ϕI (K))
K ' # K ⊃ h(ϕI (K)) " h(ϕI (K)) h(K) K h(ϕI (K)) # 5 J 1 h(ϕI (K)) K $' ! ϕJ (K) ϕJ (K) ⊂ h(ϕI (K)) ⊂ K ⊂ h(K) " ϕJ (K) K h(ϕI (K)) h(K) ! (ϕJ (K), h(K) ϕJ (K)) > 0 h g g ◦ h−1 h(K) ϕJ (K) ∩ ϕJ (K) = ∅. ! g g(K) ∩ ϕJ (K) = g ◦ h−1 h(K) ∩ ϕJ (K) = = g ◦ h−1 ϕJ (K) ∩ ϕJ (K) g ◦ h−1 h(K) ϕJ (K) ∩ ϕJ (K) = = g ◦ h−1 ϕJ (K) ∩ ϕJ (K). & 5 4 6 # $ h g g = h μ g ◦ h−1 ϕJ (K) ∩ ϕJ (K) < c · μ ϕJ (K) = c · pJ . μ g(K) ∩ ϕJ (K) = μ (g ◦ h−1 )(ϕJ (K)) ∩ ϕJ (K) < c · pJ
μ g(K) ∩ K = μ g(K) ∩ ϕJ (K) + μ g(K) ∩ (K ϕJ (K)) < c · pJ + 1 − pJ = 1 − (1 − c)pJ . +4 " ) h ' ' c < 1 G h # g μ g(K) ∩ K < c ! /. 4 = 1 0 c < 1 g ∈ G μ g(K) ∩ K < c g(K) ⊃ K / 5 9 6 0 G ' ! K ' ϕI (K) $ & ϕI (K) $ c ' K μ g(K) ∩ ϕI (K) < c · μ ϕI (K) ϕI (K) ⊂ g(K) ,- K = ϕ1 (K) . ϕr (K) μ K & ! & g μ g(K) ∩ K > 0 g(K) ∩ K #
K & μ K (g(K) ∩ K) = μ g(K) ∩ K % % g " ! ( g(K) ∩ K # # # 2 1 c 5 = K ' 1 g ' μ g(K) ∩ K > 0 & $ 4 J 1 ε = 1 − c 1 ' $ ai (K) μ g(K) ∩ ai (K) = μ (g(K) ∩ K) ∩ ai (K) > c · μ ai (K) g(K) ∩ K i ai (K) $ 2 & 5 = ai (K) ⊂ g(K) " ai (K) K g(K) ∩ K i ai (K) ⊂ K (g(K) ∩ K) ! μ g(K) ∩ K = μ g(K) ∩ K ∩ ai (K) + μ (g(K) ∩ K) i ai (K) = ai (K) ≤ μ K (g(K) ∩ K) , =μ i ! " #
& . # 2 ' ) K = ϕ1 (K) . ϕr (K) / g # g(K) ⊂ K g(K) 0 " ! & g ' $' & . K ⊂ g−1 (K) & $' $ ' 5 9 6 # +5 & h ∈ H ' LM h(K) ∩ K = ∅ # H / 5 5 6 0 " # . ) K = ϕ1 (K) . ϕr (K) # λ(K) ⊂ K ! k ≥ 1
I, J λ 1 k λ ◦ ϕI = ϕJ " ! " k ≥ 1 ϕI (K) k λk (K) ! I (ϕ−1 I ◦ λ )(K) ⊂ K )' % ' ' ) 9 + 1 −1 −1 k k k −k λk −k ◦ ϕ = ϕ k < k I I ϕI ◦ λ = ϕI ◦ λ ϕI ◦ ϕ−1 I I I =λ / - & λ(K) ⊂ K λ ' μ(λ) μ λ(K) ( ' # )' $ ◦ g1 g2 g1 ◦ g2 gk k g # )' & μ ' 2 ) . ' ' ϕI ' μ ◦ λ(K) ⊂ K $ λ
' μ ◦ λ = μ(λ) · μ ϕI = μ(ϕI ) · μ & ( ! K = ϕ1 (K) . ϕr (K) μ $ * ! # + # λ(K) ⊂ K μ ◦λ = μ λ(K) ·μ % μ λ(H) = μ λ(K) · μ H λ H ⊂ K " ϕI (K) μ g(ϕI (K)) ( g # g(ϕI (K)) ⊂ K μ ϕI (K) = " ! & λ ' # # & ' $ & 9 + 1 ) λ ' λ(K) ⊂ K λ(K) )' - λ0 , λ1 , . , λt
λ0 & $ ' F' λ ' λ(K) ⊂ K ϕI ◦ λi I i /$ ϕI (K) λ(K) ϕ−1 I ◦ λ ' K # )' +9 λi 0 & # μ ◦ ϕJ = pJ · μ = μ(ϕJ (K)) · μ J $ M H ⊂ K 3 μ λ(H) = μ (ϕI ◦ λi )(H) = μ ϕI (K) · μ λi (H) = μ ϕI (K) · μ λi (K) · μ H = = μ (ϕI ◦ λi )(K) · μ H = μ λ(K) · μ H , $ " 0 ≤ i ≤ t 9 4 1 Ii Ji $ M ki # λki i ◦ ϕIi = ϕJi 1 bi = ϕIi ci = ϕJi λki i bi = ci
1 μ∗ (λi ) = ki μ(ci ) . μ(bi ) 7$ μ∗ (λi ) = μ(λi ) " 0 ≤ i ≤ t I $ M 0 ≤ j ≤ t J $ M λi ◦ϕI = ϕJ ◦λj / ϕJ (K) (λi ◦ϕI )(K)0 ( ' # ))'; g1 g2 ' g1 ≈ g2 g1 ◦ g2−1 ' H g1 (H) g2 (H) ' g1 g2 . ! ' . # g1 ◦ g2 ≈ g3 ⇐⇒ g2 ◦ g1 ≈ g3 k $ λi ϕI = ϕJ λj λki i bi = ci λj j bj = cj ' kk kk k k k k kk k k k λi i j ϕI i j bi j bkj i ≈ ϕJi j bi j λj i j bkj i ≈ ϕJi j bi j ckj i , k k k k k k k k k ≈ϕJi j λj i j ≈cj i k kk k k k k kk λi i j ϕI i j bi j bkj i ≈ λi i j bi
j ϕI i j bkj i ≈ ci j ϕI i j bkj i . k ≈ci j & kk k k k k ϕJi j bi j ckj i ≈ ci j ϕI i j bkj i . & bi bj ci cj . ' ' # )' ' k k k k k k ϕJi j bi j ckj i (K) ci j ϕI i j bkj i (K) ' . & μ(ϕJ )ki kj μ(bi )kj μ(cj )ki = μ(ci )kj μ(ϕI )ki kj μ(bj )ki , μ∗ L)' # μ(ϕJ )ki kj μ∗ (λj )ki kj = μ∗ (λi )ki kj μ(ϕI )ki kj , μ∗ (λj )μ(ϕJ ) = μ∗ (λi )μ(ϕI ). μ∗ (λi )μ ϕI μ λj . μ(λi ϕI ) = μ(ϕJ λj ) = μ(ϕJ )μ(λj ) = μ∗ λj +: ! ) ; i I j μ(λi ϕI ) = μ∗ (λi ) μ(λj ) μ(ϕI ). μ∗ (λj ) ! μ∗ (λj ) = 0 L)' . 1 m M μ(λm )
˙ μ(λi ) ≤ ∗ μ∗ (λm ) μ (λi ) 0 ≤ i ≤ t M Q # $ $ $ ! ϕI μ(λm ϕI ) = μ∗ (λm ) μ(λj ) ˙ μ∗ (λm ) μ(λm ) μ(ϕI ) = μ(λm )μ(ϕI ) μ(ϕI ) ≥ ∗ μ (λj ) μ∗ (λm ) 0 ≤ j ≤ t 1 {ϕIi (K)} ' ' #)' K ϕI (K) # ! μ λm (K) = μ λm ϕIi (K) ˙ μ λm ϕIi ≥ ˙ ≥ μ(λm )μ(ϕIi ) = μ(λm ), =μ λm (ϕIi (K)) = μ(λm ϕI ) = μ(λm )μ(ϕI ) I $ M 1 H ⊂ K 3 & μ L)' . aij (K) H ⊂ j i aij (K) μ(H) = inf j μ i aij (K) = μ j i aij (K) ! μ(λm (H)) ≤ μ λm j i j = μ(λm ) inf j j i R H c = K H =μ λm (aij (K)) = inf ≤ inf μ j aij (K)
λm (aij (K)) ≤ i μ(λm aij ) = inf j i μ(aij ) = μ(λm )μ i i j i μ(λm )μ(aij ) = aij (K) = μ(λm )μ(H). μ λm (H c ) ≤ μ(λm )μ(H c ) & μ λm (H)) + μ λm (H c ) ≤ μ(λm )μ(H) + μ(λm )μ(H c ) ' H μ λm (H) = μ(λm )μ(H) μ ◦ λm = μ(λm ) · μ ! # $ H ⊂ K 3 n−1 n−1 μ λnm (H) = μ λm (λm (H)) = μ(λm )μ λm (H) , $)' μ λnm (H) = μ(λm )n μ(H) #) μ(λnm ) = μ(λm )n ! μ(λkmm bm ) = μ(λm )km μ(bm ) & μ∗ (λm ) L)' μ∗ (λm )km μ(bm ) cm = λkmm bm μ(cm ) μ(λm )km μ(bm ) = μ(λkmm bm ) = μ(cm ) = μ∗ (λm )km μ(bm ). +J = i) " μ(bm ) > 0 # $ μ(λm ) = μ∗ (λm ) " m i M μμ(λ ∗ (λ ) i 0 ≤ i ≤ t μ∗
(λi ) ≤˙ μ(λi ) m) & m M μμ(λ ∗ (λ ) ≥ m μ(λi ) 0 ≤ i ≤ t ) # . $ ∗ μ (λi ) # $ 0 ≤ i ≤ t μ∗ (λi ) ≥ μ(λi ) ! 0 ≤ i ≤ t μ∗ (λi ) = μ(λi ) ! m ' 0 ≤ i ≤ t $ i μ ◦ λi = μ(λi ) · μ & λ(K) ⊂ K λ ' μ ◦ λ = μ(λ) · μ μ ◦ λn = μ(λ)n · μ " # . ϕL (K) g ' K # g(ϕL (K)) ⊂ K & 9 4 1 I J $ M k # (g ◦ ϕL )k ◦ ϕI = ϕJ ! # μ(ϕJ ) = μ (g ◦ ϕL )k ◦ ϕI = μ(g ◦
ϕL )k μ(ϕI ), ϕJ = (g ◦ ϕL )k ◦ ϕI ≈ (ϕL )k ϕI μ(ϕJ ) = μ((ϕL )k ϕI ) = μ(ϕL )k μ(ϕI ), μ(g ◦ ϕL )k μ(ϕI ) = μ(ϕL )k μ(ϕI ), μ(g ◦ ϕL ) = μ(ϕL ), $ . 0 ' 1 2 K = ϕ1 (K) . ϕr (K) # # μ $ ! μ # & K & " ! & $ g ' H ⊂ K 3 g(H) ⊂ K μ(H) = μ(g(H)) 1 c 5 = c < 1 ! H ⊂ K # 2 & $ 4 : 1 ε = 1−c H # $ a(K) μ H ∩ a(K) > c · μ a(K) " H ⊂ g−1 (K) μ g−1 (K) ∩ a(K) > c · μ a(K)
5 = a(K) ⊂ g−1 (K) g(a(K)) ⊂ K 1 λ = g ◦ a & 9 5 6 # μ(λ) = μ(a) / μ(λ) μ λ(K) 0 H0 = a−1 (a(K) ∩ H) μ(λ(H0 )) = μ(λ)μ(H0 ) 0 < c · μ a(K) < μ a(K) ∩ H = μ a(H0 ) = μ(a)μ(H0 ) = μ(λ)μ(H0 ) = = μ λ(H0 ) = μ g(a(H0 )) = μ g(a(K) ∩ H) ≤ μ g(H) , += g(H) # 2 F # 2 ' # # 2 ## $ 2 ' # $ 2 1 H ⊂ K 3 g # ' g(H) ⊂ K & $ 4 J 1 0 < ε < 1 − c ! ai (K) μ H ∩ ai (K) > (1 − ε) · μ ai (K) μ H ai (K) = 0. i ! H ⊂ g−1 (K) μ g−1 (K) ∩ ai (K) > (1 − ε) · μ ai (K) & 5 = # g−1
(K) ⊃ ai (K) g(ai (K)) ⊂ K & 9 5 6 μ g(ai (K)) = μ ai (K) . $ 2 ' # $ 2 μ g(H) = μ g H ∩ ai (K) +μ g H i = ai (K) =μ g H∩ i ai (K) = i μ g(H ∩ ai (K)) ≤ μ g(ai (K)) = μ ai (K) ≤ i i 1 1 · ·μ H ∩ μ H ∩ ai (K) = ≤ 1−ε 1−ε i i ai (K) = i 1 · μ(H). 1−ε ! 0 < ε < 1 − c μ g(H) ≤ μ H R H g(H) g g−1 μ H ≤ μ g(H) μ H = μ g(H) μ ' 1 " 4 2 ' - αi ϕi ' F # ' K ' # μ
# ' αi1 αi2 . αin = αj1 αj2 αjm pi1 pi2 . pin = pj1 pj2 pjm μ pi & αi pi $ #$ − log pi ' − log αi (i = 1, . , r) # ' . ) Q . #) r . Q ' & ' − log pi ' # r ' Rr # ' / ' 0 i pi = 1 . # ( 1 ' # 1 # ($ 8 $ $ /%
' pi = αsi $ $ 0 & . ' & ' +< . K μ # &# % # 2 {ϕ1 , ϕ2 , . , ϕr } # &# ! μ " ! 1 {ψ1 , . , ψs } & 9 4 1 k # ϕI ϕJ ψik ◦ ϕI = ϕJ 9 5 6 # # 0 < μ(ϕJ ) = μ(ψik ◦ ϕI ) = μ(ψi )k μ(ϕI ), μ(ψi ) > 0 1 ≤ i ≤ s & 9 5 6 # μ ◦ ψi = μ(ψi ) · μ μ ψi (K) = 1 μ {ψ1 , . ,
ψs } # & )' 2 * ,- K = ϕ1 (K) . ϕr (K) μ # ! 0 # μ # & & # " ! & 9 9 $ ' 1 ϕI (K) ϕJ (K) ' ! ϕ2I (K) ϕ2J (K) ' ϕ2I ϕ2J ' ϕ2I ◦ ϕ−2 J . 2 2 2 μ(ϕI (K)) = μ(ϕJ (K)) & μ(ϕI (K)) = μ(ϕI (K))2 μ(ϕ2J (K)) = μ(ϕJ (K))2 μ(ϕI (K)) = μ(ϕJ (K)) ! $ $ % ! $$ ' / $ '0
) . 1 ϕ1 (x) = x3 ϕ2 (x) = x3 + 23 K # K = ϕ1 (K) ∪ ϕ2 (K) 7 1 ψ1 (x) = − x3 + 13 ! K = ψ1 (K) ϕ2 (K) . ) ' ' ! . O P; {ϕ1 (K), ϕ2 (K)} {ψ1 (K), ϕ2 (K)} ' )' {a1 (K), . , ar (K)} {b1 (K), . , bs (K)} j = 1, s 1 ≤ i ≤ r bj (K) ⊂ ai (K) ! . $ ' ) $ 2 7 . +@ & $ . $
' ) . K ' ; ( λ1 λ2 ' λ1 (K) ⊂ K λ2 (K) ⊂ K λ1 (K) ∩ λ2 (K) = ∅ λ1 (K) ⊂ λ2 (K) λ2 (K) ⊂ λ1 (K) & 5 2 $ ' ' ' & $)' $ " . K ) . 1 a, b, c # ' a + b + a + c + a + b + a = 1 b = a · c ! 2 0 < a < 1/4 b c . 1 ϕ1 [0, 1] [0, a] # ' ' F ϕ2 [0, 1] [a+b, a+b+a] ϕ3
[1−a−b−a, 1−a−b] ϕ4 # [1−a, 1] ' F ϕ1 (x) = a·x ϕ2 (x) = a·x+a+b ϕ3 (x) = a · x + 1 − a − b − a ϕ4 (x) = a · x + 1 − a D + ϕ1 ([0, 1]) ϕ2 ([0, 1]) ) ψ1 ([0, 1]) ϕ3 ([0, 1]) ϕ4 ([0, 1]) ψ2 ([0, 1]) 1 K # K = ϕ1 (K)ϕ2 (K)ϕ3 (K)ϕ4 (K) ! K )' a 2 b c $ F K ⊂ [0, 1] K $ 12 & K . # ϕi ([0, 1]) [0, 1] ' ' /a = 0, 15 c = 0,4 1,3 0 & & ϕi (ϕj ([0, 1])) (1 ≤ i, j ≤ 4) . & K #'$ ' 1 ψ1 (x) = a · x + a2 + a · b + a2 + a · c ψ2 (x) = a · x + 1 − a − b
− a2 − a · b − a2 & [0, 1] ϕ21 ϕ1 ◦ ϕ2 ψ1 ϕ2 ◦ ϕ3 ϕ2 ◦ ϕ4 ϕ3 ◦ ϕ1 ϕ3 ◦ ϕ2 ψ2 ϕ4 ◦ ϕ3 ϕ24 ' # . & $ ψ1 (K) ⊂ K ψ2 (K) ⊂ K {ϕ21 , ϕ1 ◦ ϕ2 , ψ1 , ϕ2 ◦ ϕ3 , ϕ2 ◦ ϕ4 , ϕ3 ◦ ϕ1 , ϕ3 ◦ ϕ2 , ψ2 , ϕ4 ◦ ϕ3 , ϕ24 } K ' . / 0 ! $ ψ1 ◦ ϕ1 = ϕ1 ◦ ϕ3 ψ1 ◦ ϕ2 = ϕ1 ◦ ϕ4 ψ1 ◦ ϕ3 = ϕ2 ◦ ϕ1 ψ1 ◦ ϕ4 = ϕ2 ◦ ϕ2 4D ψ2 ◦ ϕ1 = ϕ3 ◦ ϕ3 ψ2 ◦ ϕ2 = ϕ3 ◦ ϕ4 ψ2 ◦ ϕ3 = ϕ4 ◦ ϕ1 ψ2 ◦ ϕ4 = ϕ4 ◦ ϕ2 ! 2 b = a · c " $ $ ) { 1 (K), . , r (K)} K ' K {ϕ1 , ϕ2 , ϕ3 , ϕ4 } . L .
' ! K 2 . /F' 0 % . { 1 (K), , r (K)} " ϕ1 (K) ∩ ψ1 (K) = ∅ i ϕ1 (K) ∪ ψ1 (K) ⊂ i (K) R i ϕ2 (K) ∪ ψ1 (K) ⊂ i (K) ( ' j M ϕ3 (K) ∪ ψ2 (K) ∪ ϕ4 (K) ⊂ j (K) ! K = i (K) j (K) i j ' ) [0, 1] [0, a + b + a] [1 − a − b − a, 1] # b = (a + b + a) · c . b = a · c & λ1 λ2 ' # $ ϕ1 ψ1 ' . & *+, - ! ($) N) : =+5S=9= *4, T$ E 3 2 ' ( . 3 # ( 4 3 /+@<+0 . $ ) 0 3
/+@@<0 +D 9DJ:S9D<= . 7 *5, 7 C$ & " -7 # T )) ) . $ 7 B ($ 8 ' ( 5+6 /4DD+0 + 9:S@J *9, N Q -$ 1 % ) . ) 7 B % N$ # # 7 80 /4DD40 @ +<D=S+<+= *:, UM 1 I V % ) . ( 0 5! ! 6 # /+@@@0 4 4+9S4+@ . 7 ) '0 *J, G H I B ) $ L . $ ( /+@=+0 +J+S+J4 *=, ! " 3 B ) $ σ L . $ '# ( *<, 3 . E I R . UB) +@<9 *@, 7$ ")"$ ($ #* /+@<90 +S@ 8 .
#A )# 4+ *+D, I C $ E 1 ($ 8 M .? ) /+@@40 :55S:J< *++, T$ E #) $ . ? )# ! /+@@90 5 95=S9:D . ) *+4, T$ E .? )# ")"$ 3 L ($ $ ! * /+@@90 :+5S:4J *+5, 1) ) "' / %# !1! +@@: 44 8
1817. március 2-án született Nagyszalontán. Tizenhat éves koráig otthon nevelkedett; előbb szülei oktatták féltő gonddal kései gyermeküket, majd a helyi iskolában tanult és segédtanítóskodott. 1833 novemberében a debreceni kollégium főiskolai tagozatára került, „tógátus” diák lett, de egy év múlva anyagi nehézségek miatt megszakította tanulmányait; 1834
