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?

  • 3
    $\begingroup$ Measurable function needs not have finite integral, while integrable does by definition. $\endgroup$ – 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.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.