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## 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?