請教兩題:
P.70 第21題.b
題目:Let f be a function with measurable domain D. Show that:
f is measurable <==> g(x) is measurable. g(x) = { f(x) ,for x belong to D
{ 0 ,for x NOT belong to D
解:
==>: Suppose f is measurable.
case 1. α≧ 0, {x:g(x)＞α} = {x:f(x)＞α} , which is measurable.
_
case 2. α＜ 0, {x:g(x)＞α} = {x:f(x)＞α} ∪ D, which is measurable.
_
不懂 D (D的complement) 為何是measurable???
P.66 第17題.b
題目: Give an example of a decreasing sequence of sets <E_i>, m*(E_i)＜∞,
and m*(∩E_i) ＜ lim (m*E_i) .
解:
Let E_i = ∪ P_n. If x belong to P_k, then x not belong to ∪ P_n,
n≧i n≧k+1
So, ∩(E_i) = ψ,so m*(∩(E_i)) = 0.
On the other hand, P_i 被包含於 E_i, for each i.
So m*(E_i) ≧ m*(P_i) = m*(P) ＞ 0, for each i.
不懂m*(P)＞ 0??
我的理解是 m*(P) 只有2種情況,分別是 m*(P) 等於 0 或等於 ∞.
這邊說 m*(P)＞ 0,是否即指 m*(P)等於 ∞ ???
還有我們又如何知道 m*(P) ≠ 0 呢??
感謝回答!
--
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 58.115.75.19