Re: [分析] 實變三題

[cgkm (cgkm)]

※ 引述《icebergvodka (肥嘟嘟左衛門)》之銘言：
: 1.
:   If u is an outer measure and if every open set is measuable,
: then u is a metric outer measure.

Proof. Say d(A, B) > 0. X\cl(A) is open, hence measurable,
so cl(A) is measurable as well, and
        A ∪ B ∩ cl(A) = A
        (A ∪ B) \ cl(A) = B
hence
        u(A ∪ B) = u(A ∪ B ∩ cl(A)) + u((A ∪ B) \ cl(A))
                  = u(A) + u(B). □

: 2.
:   Consider the transformation Tx=Ax+k in R^n, where A is a nonsingular
: n*n matrix and x, k are column n-vectors. T maps sets E onto sets T(E).
: Assume that 入[T(I(a,b))] = |detA|入(I(a,b)). Prove that satisfies the
: properties (a)-(c).
: (a) For any set E, u(T(E))=|detA|u(E), u is outer measure.
: (b) E is lebesgue-measurable if and only if T(E) is lebesgue-measurable.
: (c) If E is lebesgue-measurable, then u(T(E))=|detA|u(E), u is lebesgue
: measure.

Hints:
(a) holds on open rectangles, so consider countable covers for T(E)
    of open rectangles, etc.
(b) holds because T is a homemorphism (i.e. T is a continuous bijection
    whose inverse is also continuous).
(c) follows from (a) and (b).

: 3.
:   Given an example of a signd measure for which the Hahn decomposition is not
: unique.

#1AsVzImI has an example.

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◆ From: 75.62.141.216
※ 編輯: cgkm            來自: 75.62.141.216        (11/25 16:54)
推 icebergvodka:感激解惑~@@
→ icebergvodka:我解釋一下I(a,b)的定義
→ icebergvodka:I(a,b)={x=(x_1,x_2,...x_n) | a_i<x_i<b_i for i=1,
→ icebergvodka:2..n}
→ icebergvodka:其實我是看不太懂課本上對I(a,b)的定義..想請問一下
→ icebergvodka:這定義其實就是cgkm大所假設的樣子嗎??
是的（但我現在發覺，我那個「(a , ..., a ) × (b , ..., b )」
                              1        b       1        n
的寫法好像是我自己亂編的，所以我把那幾行刪掉了，以下重新說明）

If a  ＜ b  for all i, then
    i     i
                          n
        I(a, b) = {x in Ｒ : a  ＜ x  ＜ b  for all i}
                              i     i     i

                = (a , b ) × ... × (a , b )   (Cartesian products)
                    1   1              n   n

                           n
is the open rectangle in Ｒ  with corners a and b.

                                       2
For example, the open unit square in Ｒ  is
        I((0, 0), (1, 1))
                              n
and the open unit square in Ｒ  is
        I((0, ..., 0), (1, ..., 1)).


Open recangles in Ｒ are open intervals.

The sides of an open rectangle have I(a, b) lengths b  - a ,
                                                     i    i
so the Lebesgue measure of I(a, b) is
        Π (b  - a ) = (b - a ) ... (b - a ).
             i    i      1   1        n   n
※ 編輯: cgkm            來自: 75.62.141.216        (11/26 01:35)
※ 編輯: cgkm            來自: 75.62.141.216        (11/26 05:54)
※ 編輯: cgkm            來自: 75.62.141.216        (11/26 07:50)
推 math1209    :homemorphism 只能將 Borel set 送到 Borel set.
→ math1209    :(b) 應該使用 Lipschitz 函數會將可測集送至可測集.