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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?

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

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