In Lebesgue integration, why is it so important to have properties usually true almost everywhere ? Is it because a function like $1_{\mathbb{Q}}$ is not integrable with Riemann integration ? I am not satisfied by this answer... A clarification is welcome.
Thank you so much.
Marcus
The reason is that with Lebasgue integration, subtracting a set of measure 0 has no effect on the integral. Therefore we don't NEED properties to hold everywhere, it's good enough for them to hold on all but a measure 0 set, since then the integral will be the same.