Thread: proof an inequality; measure theory

1. proof an inequality; measure theory

Hi there

Let C be compakt
$\displaystyle \mu$ a radon measure and $\displaystyle \mu(C) < \infty$.

Let $\displaystyle v \in L^1(\mathbb{R}) , \ 0 < a < b < \infty$

Do you know how to proof the following inequality:

$\displaystyle \int_{a}^b |v| d\mu \le \mu([a,b]) \cdot \|v\|_\infty$

I don't get it.
By the way I'm not sure about the correctness of this inequaility if $\displaystyle \mu$ is an arbitrary measure. If it's correct where $\displaystyle \mu$ equals the Lebesgue measure, you can use this instead of the Radon measure.

Yours Rapha

2. Presumably $\displaystyle \mu$ is supposed to be a positive Radon measure.

From the definition of the $\displaystyle L^\infty$-norm, $\displaystyle |v(t)|\leqslant \|v\|_\infty$ for almost all t in [a,b]. Then $\displaystyle \int_a^b\!\!\!|v(t)|\,d\mu(t) \leqslant \int_a^b\!\!\!\|v\|_\infty\,d\mu(t) = \mu([a,b]) \|v\|_\infty$ (integral of a constant is the constant times the measure of the interval).

3. Hello Opalg

thank you very much!

Kind regards
Rapha