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examples of measurable functions on $\mathbb{R}$

Is there a measurable function $g:\mathbb{R} \to \mathbb{R} ^{+}$ such that $\int_{[a,b]}g dx=\infty$ $\forall a\neq b \in\mathbb{R}$ .

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    Are we assuming Lebesgue measure?2012-11-22
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    yes I will edit it2012-11-22
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    Clearly g cannot be continuous...2012-11-22
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    Yes otherwise it will have a maximum by the extreme value theorem2012-11-22
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    See: http://math.stackexchange.com/q/244132012-11-22

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