The definition of integration that I am using: The predicate $CSM(p)$ is a shorthand for $p$ having a countable range and $p$ is a measurable function. Let $p:X\rightarrow [0,\infty]$ be a function such that $CSM(p)$. We will define the integral of $p$ to be: $\int_X p\ d\mu=\sum_{a\in p(X)}a\mu(p^{-1}(\{a\}))$ Now let $f:X\rightarrow [0,\infty]$ be any measurable function, we define the upper and lower integrals of $f$ as:
$\int_{X}^{*}f\ d\mu=\inf\{\int_{X} p \ d\mu|p:X\rightarrow[0,\infty],CSM(p),\mu(\{x\in X|f(x)>p(x)\}=0\}$ $\int_{*X}^{}f\ d\mu=\sup\{\int_{X} q \ d\mu|q:X\rightarrow[0,\infty],CSM(q),\mu(\{x\in X|f(x)
Finally, we say that the integral of $f$ exists iff $\int_{X}^{*} f\ d\mu=\int_{*X}^{}f\ d\mu$ and we denote it $\int_X f\ d\mu$.
To prove your theorem Verify that if $f=g$ $\mu.a.e$ then:
1) For all measurable functions $p$ , $f\leq p$ a.e. iff $g\leq p$ a.e.
2) For all measurable functions $q$ , $f\geq q$ a.e. iff $g\geq q$ a.e.
Finally deduce that the upper and lower integrals of $f,g$ are equal.