# Basic Measure Theory Question

1. May 8, 2012

### EV33

1. The problem statement, all variables and given/known data
My question is would I be allowed to say,
if lf+-$\phi$l<ε/(2$\mu$(E)
then ∫E lf+-$\phi$l<ε/2

2. Relevant equations

E is the set in which we are integrating over.
$\mu$ is the measure
$\varphi$ is a simple function
f+ is the non-negative part of the function f.

3. The attempt at a solution

I can't think of any reason that this wouldn't be true but my text is very vague in this chapter and so I am really not sure if this is an ok statement.

Thank you for your time.

2. May 8, 2012

### Fredrik

Staff Emeritus
Yes, your conclusion is correct.

It's not hard to prove the following:
$\newcommand{dmu}{\operatorname{d}\!\mu}$

(a) If $f\geq 0$ a.e., then $\int f\dmu \geq 0$.
(b) If $f\geq g\geq 0$ on E, then $\int_E f\dmu\geq\int_E g\dmu$.

Hint: To prove (b), use (a) and the fact that the assumption implies that $f\chi_E\geq g\chi_E$ (everywhere, and therefore a.e.).