2022-07-10

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Let $a\in X$ and $\mu$ be a measure defined on ${2}^{X}$ by

Write a simple necessary and sufficient condition on the non-negative functions $f$ that ensures that

My attempt:
We know that , so we want the supremum to be finite and for this I'm thinking that the functions should have a finite border but I'm not sure if it works!

Tanner Hamilton

Expert

$\sum {y}_{i}\mu \left({A}_{i}\right)={y}_{i}$ if there $i$ is such that $a\in {A}_{i}$ and 0 if there is no such $i$. Note that if ${A}_{i}$'s are disjoint there can be at most one i for which $A\in {A}_{i}$ Hnece $\int sd\mu =s\left(a\right)$. From this it follows that $\int fd\mu =f\left(a\right)$ for all non-negative measurable functiosn $f$. Hence, the conditioin is $f\left(a\right)<\mathrm{\infty }$.