Let $~f:[0,1] \to[0,\infty]$ be a measurable function bounded by $c \in \mathbb R$. Let $X_1,X_2,..,X_n \sim i.i.d ~\text{uniform}(0,1)$.
How do I interpret the following statement: $$ Var(f \circ X_1)\le c^2 $$ Does it just mean that $Var (f(X_1)) \le c^2$? and how is this true anyway? Is it proved in the following: \begin{equation*} f(x) \le c \implies f^2(x) \le c^2 \end{equation*} and so \begin{equation*} E(f^2(x)) \le E[c^2]=c^2 \implies Var(f(x) )=E[f^2(x)]-E[f(x)]^2 \le c^2 \end{equation*}