Question about integrable functions

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Suppose you have a nonnegative Lebesgue measurable function supported on [tex][0,1][/tex] such that [tex]\int_0^1 f = \infty[/tex]. Does this mean that the set [tex]E = \{ x\in [0,1] : f(x) = \infty \}[/tex] has positive Lebesgue measure?
 
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No. Consider the function [itex]f\left(x\right)=1/x[/itex], and whatever value you want at x = 0.
 

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