Let $(X, \mathcal B, \mu)$ be a finite measure space. Prove that $L^2(X, \mathcal B, \mu) \subset L^1(X, \mathcal B, \mu)$.

I will be glad for any idea, comment, hint or advice.

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# $L^2\subset L^1$ in case if the measure is finite

###### Related Question

functional-analysislebesgue-integralmeasure-theoryreal-analysis

Let $(X, \mathcal B, \mu)$ be a finite measure space. Prove that $L^2(X, \mathcal B, \mu) \subset L^1(X, \mathcal B, \mu)$.

I will be glad for any idea, comment, hint or advice.

## Best Answer

Hint: Let $f: X \to \mathbb{R}$ be non-negative and in $L^2$. Separate $f = f|_{\{f\le 1\}}+f|_{\{f> 1\}}$.

For the first part, notice that the measure space is finite. For the second part, notice that for some (which ones?) $\mathbb{R}\ni a>0$ we have $a \le a^2$.