What does it mean to say a measurable function is integrable with respect to a measure, such as $\mu$? I know the definition of integrability, but I'm still not sure what exactly what "with respect to a measure" means.
For example, with Riemann integration, for a function of one variable, $f(x)$ say, we know we are integrating with respect to $x$ and $x$ is clearly in the integrand. How should I interpret $d\mu$? Or $d\mu(f^{-1}(B))$?
This question is to help me understand Radon Nikodym derivatives.
Best Answer
A measurable function is integrable with respect to the measure $\mu$ if
$$\int_\Omega |f(x)| d\mu < \infty$$
If you change the measure, you change which functions are integrable.
For exemple if $\mu$ is the dirac in 0, then every function is integrable (assuming a function at value in $\Bbb R$)