Why Does a Zero Integral Imply a Function Is Zero Almost Everywhere?

[Ja4Coltrane]

Homework Statement

Let (X,m) be a measure space, f:X->[0, infinity] be measurable.

If the integral over X of f is 0, show that f=0 almost everywhere

Homework Equations

The Attempt at a Solution

Suppose that f is nonzero on A, m(A)>0.

I've reduced problem to proving that the following is impossible:

f is positive on a set of nonzero measure and
{x such that f(x)>epsilon} has zero measure for all epsilon greater than zero.

I don't know how to do that though...

Any help?

 

[HallsofIvy]

Suppose the statement is not true- that there exist a set A, of measure greater than 0, on which f is positive. Then the integral of f over A alone is positive and, since f is never negative, the integral of f over X is not less than the integral of f over A.