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Let $\mu$ denote a finite Borel measure on $\mathbb{R}$. What are the following limits: $\lim_{t\rightarrow\infty}\int_{\mathbb{R}}f(tx)d\mu(x)$ and $\lim_{t\rightarrow0}\int_{\mathbb{R}}f(tx)d\mu(x)?$ $f$ is a continuous function on $\mathbb{R}$ with compact support here.

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What do you think they are? On what do you think they will depend? –  t.b. Aug 6 '11 at 9:23

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If "finite measure" means $\mu(\mathbb{R})<\infty$ or even if it means the measure of the compact support of $f$ is finite, then, since continuity of $f$ on a compact set implies boundedness of $f$, the dominated convergence theorem should be applicable. Since $f$ is continuous, its limit at 0 will be $f(0)$, a constant that can be pulled out from under the integral. The fact that $f$ is continuous and has compact support tells you what its limit at $\infty$ is.

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