if we have a function $f \in L^p$ sucht that $||f||_p =1$ and $m$ being a finite measure.

Define a new measure $\mu$ by

$$\mu(A):=\int_A |f(x)|^p dm(x).$$

Then $\forall \epsilon > 0 \ \ \exists \delta>0$ such that the following implication holds:

$$m(A)< \delta \Rightarrow \mu(A) < \epsilon.$$

My question is: How is this property called or does it not have a name?


Such a measure is called absolutely continuous [with respect to the Lebesgue measure].


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy