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Let $(X,\mathcal{F},\mu)$ be a measure space. Is there a $\mu$-measurable function $f$ which satisfies the following properties?

0) $f \geq 0$

1) $\int_X f d\mu < \infty$

2) The function is not identically zero

3) $\inf \{f(x) | x \in F \} = 0$, $\forall F \in \mathcal{F}_\mu, F \neq \phi$

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up vote 4 down vote accepted

There is no measurable function satisfying (0), (2), and (3). If $f$ is a nonnegative measurable function that is not identically zero, then for each $\lambda>0$, the set $F_\lambda=\{x\in X:f(x)>\lambda\}$ is measurable, and since $\displaystyle{\{x\in X:f(x)>0\}=\cup_{\lambda>0} F_\lambda}$, not all $F_\lambda$s are empty. Let $\lambda_0>0$ be such that $F_{\lambda_0}\neq \emptyset$. Then $\inf\{f(x):x\in F_{\lambda_0}\}\geq\lambda_0>0$.

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Thanks, nice answer. These were the three important conditions for me. I can do away with (1). – jpv May 26 '11 at 16:18

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