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Let $\mu(\cdot)$ be a probability measure on $X \subseteq \mathbb{R}$.

Consider a measurable but not integrable function $F: X \rightarrow \mathbb{R}_{\geq 0}$.

(2) Prove that there exists a function $\phi: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ such that:

(2a) $\phi(0)=0$, $\phi$ continuous and strictly increasing, $\lim_{x \rightarrow \infty} \phi(x) = +\infty$;

(2b) \int_X \phi( F(x) ) \ \mu(dx) \ < \ +\infty

EDIT:

(1) Find an example in which there is no function $f: X \rightarrow \mathbb{R}_{\geq 0}$

($f(0)=0$, $f$ continuous and strictly increasing, $\lim_{x \rightarrow \infty} f(x) = +\infty$) such that:

\int_X f(x) F(x) \ \mu(dx) \ < \ +\infty

  • 0
    If what you call *the main question* is part (2), then this is a duplicate.2012-04-22

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