Let $f$ be a measurable function on $[0,1]$ such that $\int_0^1 f(x)\,dx<\infty$ Let $U_1,\dots$ be a sequence of i.i.d Uniform $(0,1)$

Question

I have problems with this sequence, don't know how to start. Could anyone give me a hint, please? Let $f$ be a measurable function on $[0,1]$ such that $\int_0^1 f(x) \, dx < \infty$ Let $U_1,\dots$ be a sequence of i.i.d Uniform $(0,1).$ Define $$I_n=\frac{f(U_1)+\cdots+f(U_n)} n.$$ Show that $I_n \overset{\mathbb{P}}\to\int_0^1 f(x) \, dx$

Answer 1

The $f(U_i)$ are i.i.d with finite expected value $E[f(U_1)] = \int_0^1 f(x) dx$ so by the strong law of large numbers $\frac{f(U_1) + \dots + f(U_n)}{n}$ converges to $E[f(U_1)] = \int_0^1 f(x) dx$ almost surely and thus in probablity.