# Difference between non negative integrable functions and non negative measurable functions

Is there a difference between non negative integrable functions and non negative measurable functions?

In my notes it says

For a non negative measurable function $f: X \to \mathbb{R}$

$\int f d \mu = sup \{ \int g : g$ is a simple function and g $\leq f$ a.e $\}$

For a non negative integrable function $f: X \to \mathbb{R}$

$\int f d \mu = sup \{ \int g : g$ is a simple function and g $\leq f$ a.e $\}$

So is there a difference?

• Measurable function needs not have finite integral, while integrable does by definition. – Ennar Dec 22 '15 at 11:30

The set of integrable functions is a subset of the set of measurable functions. We cannot define the integral of a non-negative function $f$ unless $f$ is measurable, and we cannot define the integral of a measurable function $f$ (possibly negative) unless $f$ is integrable. The Lebesgue integral is usually developed as follows:

1. Define the integral for simple, non-negative functions (which by definition are measurable).
2. Define the integral for measurable, non-negative functions as you have.
3. Then introduce the concept of integrability: a measurable function is integrable if $\int |f|\ d\mu = \int f^{+}\ d\mu + \int f^{-}\ d\mu < \infty$.
4. Then we can define the Lebesgue integral for general measurable functions (possibly taking negative values) as long as they are integrable: $\int f\ d\mu = \int f^{+}\ d\mu - \int f^{-}\ d\mu$. This definition is well defined when $f$ is integrable because $f$ integrable implies $f^{+}$ and $f^{-}$ integrable.