Re: [分析] Borel function

[cgkm (cgkm)]

※ 引述《k6416337 (↖煞气a光希↘)》之銘言：
: 我想問幾個問題
: (1)Borel function f的定義是指定義在Borel set上而且{f>a} for all a都是Borel set嗎?

The following are equivalent terms:
        'Borel function'
        'Borel-measurable function'
        'function measurable with respect to the Borel σ-algebra'

Since the intervals (a, ∞) generate the Borel σ-algebra
on Ｒ, a function f is a Borel function on Ｒ if and only if

        {x : f(x) > a}

is Borel-measurable for each a.

(However, other types of intervals also generate
 the Borel σ-algebra on Ｒ.)

: (2)Borel set定義是指在Borel sigma-algebra裡面的那些集合嗎?

        Yes.

: (3)f:|R->|R is a differentiable function.Prove that the derivative f' is a
:    Borel function.
: 請問第三個的證明我該從什麼方向去著手?如果(1)對，我是要證明{f'>a}是可數多個開集
: 的交集還是可數多個閉集的聯集?

Hint:   f (x) = [f(x + 1/n) - f(x)]n,   x in Ｒ
         n

is Borel-measurable for each n and converges pointwise to f'.
Combine this with the fact that pointwise limits of measurable
functions are measurable.


: 有請高手指點

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◆ From: 75.62.131.87
推 k6416337    :感謝 那個f_n是否有寫錯啊?
謝謝指正！  ([ ... ]/n 改成 [ ... ]n)
※ 編輯: cgkm            來自: 75.62.131.87         (11/05 11:04)
推 k6416337    :能說一下為啥會生成σ-algebra?
Proof.

Let Ａ = the σ-algebra generated by {(a, ∞) : a in Ｒ}
    Ｂ = the Borel σ-algebra on Ｒ
       = the σ-algebra generated by the open sets in Ｒ

Since each (a, ∞) is an open set, Ａ is contained in Ｂ.

On the other hand,
1. For any b, [b, ∞) = ∩ (b - 1/n, ∞) is in Ａ.
                         n
2. If (a, b) is an open interval, then
        (a, b) = (a, ∞) ╲ [b, ∞)
   is in Ａ.
3. Each open set is a countable union of open intervals,
   and hence is in Ａ.

Since the open sets are contained in Ａ, it follows that
Ｂ is also contained in Ａ, and so Ａ = Ｂ.           □

The proofs for the collections of
* open intervals
* closed intervals
* half-open intervals (a, b] or [a, b)
* intervals of the form (-∞, b), (-∞, b], [a, ∞)
are all similar.
※ 編輯: cgkm            來自: 75.62.131.87         (11/05 16:51)
推 k6416337    :我大概懂了 謝啦