Integrating over a subset (Real Analysis) Let $0 < p < 1$ and $f: X \longrightarrow (0,\infty)$ measuarable with $\int_X f d\mu = 1$.
Prove that $\int_A f^p d\mu \leq \mu(A)^{1-p}$.
Someone please help?

It's a direct application of Hölder's inequality, using the exponents $1/p $ and $1/(1-1/(1/p))=1-p $. So
$$
\int_A f^p\,d\mu\leq\left (\int_A (f^p)^{1/p}\,d\mu\right)^p\left (\int_A1^{1/(1-p)}\,d\mu\right)^{1-p}\leq\mu (A)^{1-p}.
$$
