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In my lecture notes there is the following exercise:

"Characterize those measure spaces $(X, B, \mu)$ on which the semi-norm $\|f\| = \int_X |f| d \mu$ defined on $L^1(X) = \{ f \mid f \text{ measurable and} \int_X |f| d \mu < \infty \}$ of Lebesgue-integrable functions is a norm."

I thought that if I take $X$ to be finite and $\mu$ to be the counting measure then $\|\cdot\|_1$ is a norm. But I think the exercise asks me to use the Lebesgue measure so $\mu(X) = 0$ if $X$ is finite and my example breaks.

What's the correct answer?

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    I don't think Lebesgue-integrable means anything else than $\mu$-integrable here. This is sometimes called "integrable in the sense of Lebesgue". Also: you don't characterize anything, you only give an example. A hint: what happens if there's a nonvoid $\mu$-null set?2012-07-08
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    @DavideGiraudo The integral over the whole space, that was a typo.2012-07-08

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