lower Lebesgue sum

Overview

An important part of how to integrate a function. Closely related to the upper sum.

Definition

For a measure space $(X,\mathcal{S},\mu )$, and a measureable function $f:X\to [0,\infty ]$, and a (measurable) partition, the lower Lebesgue sum

$\mathcal{L}(f,P)={\sum }_{j=1}^m\mu (A_j){\inf }_{A_j}f$

Notes

This is technically only defined for positive functions (since the infimum doesn’t play well functions that go to $-\infty$). However, we can still take the sum of the negative parts of a function by taking the negative of the sum of the absolute value of the negative parts (better explained in the Lebesgue integral note)