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Let $\mu$ be a probability measure: $\mu(X) = \int_X \mu(dx) = 1$.

Consider a locally bounded function $f: \mathbb{R}^n \setminus \{0\} \times X \rightarrow \mathbb{R}_{> 0} \ $ such that:

  • $\exists \bar{z} \in \mathbb{R}^n \setminus \{0\} $ such that: $\int_X f(\bar{z},x) \mu(dx) < \infty $.

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Prove that:

$ \exists \delta > 0 \text{ s.t. } \ \ ||z-\bar{z}|| < \delta \ \Rightarrow \ \int_X f(z,x) \mu(dx) < \infty $

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    Please edit your post to make clear what you mean.2012-03-09

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Try $n=1$, $X = \mathbb R$ with $\mu(dx) = \frac{dx}{\pi (1+x^2)}$, $f(z,x) = (xz)^2$, $\overline{z} = 0$.

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    So just add $1$ to $f(z,x)$. And before you change the problem statement again to avoid this particular example, please think about whether you could modify the example to fit the new rules.2012-03-09