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?