• Support PF! Buy your school textbooks, materials and every day products Here!

Borel measurable functions

  • Thread starter wrldt
  • Start date
  • #1
13
0
A function f: E -> [itex]\mathbb{R}[/itex] is called Borel measurable if for all [itex]\alpha \in \mathbb{R}[/itex] the set [itex]\{x \in R : f(x) > \alpha \}[/itex] is a Borel set.

If f is a strictly increasing function, then f is Borel measurable.

Proof:

Let [itex]H=\{x \in \mathbb{R} : f(x) > \alpha \}[/itex]. I want to show that [itex](\alpha, \infty) \subset H[/itex].

My first guess is to assume that this is non-empty or else the result is trivial.

The next step, from a hint I was given, was to look at infimums (or supremums). However, I'm not sure how to proceed.
 

Answers and Replies

Related Threads on Borel measurable functions

  • Last Post
Replies
2
Views
3K
Replies
0
Views
1K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
0
Views
981
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
8
Views
1K
Replies
5
Views
2K
Replies
2
Views
984
  • Last Post
Replies
0
Views
892
  • Last Post
Replies
1
Views
2K
Top