Showing measures of random variables are equal

Question

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$.

Given random variables $f,g\colon X\rightarrow\mathbb{R}$, suppose I want to show that $\mu_f=\mu_g$. Does it suffice to show that $\mu_f(A)=\mu_g(A)$ for all sets $A$ that is a union of disjoint intervals?

Answer 1

The class of finite disjoint union of intervals (open/closed/other type) forms an algebra: it contains the empty set, it is stable under complementation and under finite intersections.

Then approximate.