Probability Question (Expectation Value Limit)

Show that for any $\phi \in C_0(\mathbb{R^n})$ (a continuous function on $\mathbb{R^n}$ which tends to 0 at $\infty$), if we set 

$\phi_\epsilon(x) = \mathbb{E}\phi(x + \epsilon Z)$

for any random variable $Z$ on $\mathbb{R^n}$, then $\phi_\epsilon(x)$$\phi(x)$ as $\epsilon$ → 0.

  • Should we interpret x as being real valued and deterministic?


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