Integral of Increasing Function Composed with Measurable Function/Corollary

Corollary to Integral of Increasing Function Composed with Measurable Function

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$ such that $p \ge 1$.

Let $f: X \to \R$ be a $p$-integrable function.

Then:

$\ds \norm f_p^p = \int_0^\infty p t^{p - 1} \map F t \rd t$

where $F$ is the survival function of $\size f$.

Proof

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Sources

• 2014: Loukas Grafakos: Classical Fourier Analysis (3rd ed.) ... (previous) ... (next): $1.1.1$: The Distribution Function