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Let $\mu$ be a probability measure on $X \subseteq \mathbb{R}^m$.

Prove that $|| a + B x ||^d$ is integrable iff $||x||^d$ is integrable:

$$ \int_{X} || a + B x ||^d \mu(dx) < \infty \ \Leftrightarrow \ \int_{X} ||x ||^d \mu(dx) < \infty $$

for any $a \in \mathbb{R}^n$, $B \in \mathbb{R}^{n \times m}$, $d \in \mathbb{R}_{> 0}$

Extend the result to any measurable (non constant) function $f(x)$:

$$ \int_{X} || a + B f(x) ||^d \mu(dx) < \infty \ \Leftrightarrow \ \int_{X} || f(x) ||^d \mu(dx) < \infty $$

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    The "for all $B$" must be part of the left side. Otherwise e.g. the $B=0$ case would not imply anything useful.2012-03-16

2 Answers 2