We separate the case in which the measure space is a probability space from the more general case because the probability case is more accessible for the general reader.
Probability-theoretic proof
Method 1:
From the definition of expectation:
However, X is a non-negative random variable thus,
From this we can derive,
From here, dividing through by allows us to see that
Method 2:
For any event , let be the indicator random variable of , that is, if occurs and otherwise.
Using this notation, we have if the event occurs, and if . Then, given ,
which is clear if we consider the two possible values of . If , then , and so . Otherwise, we have , for which and so .
Since is a monotonically increasing function, taking expectation of both sides of an inequality cannot reverse it. Therefore,
Now, using linearity of expectations, the left side of this inequality is the same as
Thus we have
and since a > 0, we can divide both sides by a.
Measure-theoretic proof
We may assume that the function is non-negative, since only its absolute value enters in the equation. Now, consider the real-valued function s on X given by
Then . By the definition of the Lebesgue integral
and since , both sides can be divided by , obtaining