# Borel measurable functions

1. Sep 17, 2011

### wrldt

A function f: E -> $\mathbb{R}$ is called Borel measurable if for all $\alpha \in \mathbb{R}$ the set $\{x \in R : f(x) > \alpha \}$ is a Borel set.

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

Proof:

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

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.