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 $