Question about integrable functions

[AxiomOfChoice]

Suppose you have a nonnegative Lebesgue measurable function supported on [0,1] such that \int_0^1 f = \infty. Does this mean that the set E = \{ x\in [0,1] : f(x) = \infty \} has positive Lebesgue measure?

 

[JSuarez]

No. Consider the function f\left(x\right)=1/x, and whatever value you want at x = 0.