N*******e
发帖数: 10 | 1 紧急求教下列问题，请各位不吝赐教, 多谢了！
1. Suppose f: R^n -> R is continuous. Prove that f(F) is Borel for every
closed set F; f(U) is Borel for every open set U; and if f is surjective,
then f(B) is Borel for every Borel set B.
2. Let B be a non-Lebesgue-measurable subset of R, and let f:R->R be defined
by
f(x)=exp(x) if x is in B, and f(x)=exp(-x) if x is not in B. Prove that
f^(-1)({x}) is Lebesgue measurable for each x in R, but f is not Lebesgue
measurable. | p***c
发帖数: 2403 | 2 随便想了一下
1. 如果F是闭，f(F)就是至多可列紧集的并
U开，U是至多可列半开半闭正方形，对每个正方形，像是可测的，就是一个区间减去几
个或开或闭的区间
如果f是双射，那很简单，如果只是满的，应该也不难
defined
【在 N*******e 的大作中提到】
: 紧急求教下列问题，请各位不吝赐教, 多谢了！
: 1. Suppose f: R^n -> R is continuous. Prove that f(F) is Borel for every
: closed set F; f(U) is Borel for every open set U; and if f is surjective,
: then f(B) is Borel for every Borel set B.
: 2. Let B be a non-Lebesgue-measurable subset of R, and let f:R->R be defined
: by
: f(x)=exp(x) if x is in B, and f(x)=exp(-x) if x is not in B. Prove that
: f^(-1)({x}) is Lebesgue measurable for each x in R, but f is not Lebesgue
: measurable.
| C*******r
发帖数: 10345 | 3 2. f^(-1)({x}) = 空集，或 {log(x)}, 或 {log(x),-log(x)}, 都是可测集。
let n=...-3,-2,-1,0,1,2,3...
存在n，such that [n,n+1)交B 是不可测集。（不然 并^(n）{[n,n+1)交B}=B 是可测
集，矛盾）。
不失一般性，假设n>0 (不然用B补集，也是不可测集）。那么[exp(n),exp(n+1)) 是
Borel集，而 f^(-1)([exp(n),exp(n+1)))=（[n,n+1)交B）并（(-n-1,n]交B补）
这得到一个不可测集，不然 （[n,n+1)交B）并（(-n-1,n]交B补）交[0,inf)
= n{[n,n+1)交B 推出 n{[n,n+1)交B 是可测集，矛盾。 |
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