Let $(X,\mathcal{F},\mu)$ be a probability space. Let $p \geq 2$.
Let $F$ be a finite set of functions of the form $f: X \rightarrow \mathbb{C}$, $\| f \|_{L^p(\mu)} \leq 1$. Let $(\Omega,\mathcal{G},\nu)$ be another probability space. Let $Y_1,\ldots,Y_n : \Omega \rightarrow F$ be i.i.d. random variables. Let $ Y = \mathbb{E}Y_j = \int_{\Omega} Y_j d\nu = \sum_{f \in F}\mathbb{P}(Y_j=f) f . $
In a paper I am reading, it is asserted that $ \mathbb{E} \left( \int_{X} \left( \frac{1}{n}\sum_{j=1}^{n} |Y_j(x) - Y(x)|^2 \right)^{p/2} d\mu(x) \right) \leq C^p. $
I can't figure out why this is true. Can someone help me out?