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Let $f$ be a measurable function on $\mathbb{R}$ satisfying $\displaystyle \int\mid f(x)\mid dm(x)< \infty$, where $m$ is the Lebesgue measure on the Borel subsets of $\mathbb{R}$. Show that the function $\displaystyle g(t)=\int_\mathbb{R}e^{itx}f(x)dm(x)$ is uniformly continuous.

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What have you tried? –  Jonas Meyer Feb 12 '11 at 18:53
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Also, it is considered rude to post questions in the imperative voice, that is, "Show that..." If this is homework, tag it as such, and you should cite the source of the problem if you can. –  Tyler Feb 12 '11 at 19:01

1 Answer 1

Hint: Continuity follows from dominated convergence. Vanishing at infinity follows from the Riemann-Lebesgue Lemma. The combination implies uniform continuity.

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