A.E. Equal Positive Measurable Functions have Equal Integrals/Proof 1

Proof
Let $N$ be the set defined by:


 * $N = \set {x \in X: \map f x \ne \map g x}$

By hypothesis, $N$ is a $\mu$-null set.

If $N = \O$, then $f = g$, trivially implying the result.

If $N \ne \O$, then by Set with Relative Complement forms Partition:


 * $X = N \cup \paren {X \setminus N}$

Now:

which establishes the result.