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Can I ask a homework question here?

Let $f$ be measurable and non-negative in $\mathbb R^d.$ Using Fubini's theorem, show that for $1 \leq p \lt \infty,$

$$\lVert f\rVert^p_p = \int^{\infty}_{0}pt^{p-1}\lambda(\{x:f(x)\gt t\}) \ dt.$$

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    Hint: Assume first that $f$ is a simple function and show that the equation holds. Then for your non-negative measurable $f$ choose a nondecreasing sequence of simple functions converging point-wise to $f$ and use monotone convergence theorem.2012-08-13

3 Answers 3