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?