Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
What do you think they are? On what do you think they will depend? – t.b. Aug 6 '11 at 9:23

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.