Proving Non-Negativity and Monotonicity of Integrals over a Measure Space

[EV33]

Homework Statement

My question is would I be allowed to say,
if lf⁺-\phil<ε/(2\mu(E)
then ∫_E lf⁺-\phil<ε/2

Homework 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.

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.

 

[Fredrik]

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.).