Suppose f(x) is a Lebesgue integrable function on [0,1],and it has a least positive period 1. Define $S_k(x)$ by $$S_k(x)={1\over{2^k}}\sum_{i=1}^{2^k} f(x+{i\over{2^k}})$$ Try to show that $$\lim_{k\to+\infty } S_k(x)=\int_{0}^{1} f(t)\, dt $$
for almost every x in[0,1]
My method:actually,I have proved that $$\lim_{k\to+\infty }\int_{0}^{1}| S_k(x)-\int_{0}^{1} f(t)\, dt|=0 $$
but when I try to use Lebesgue dominated convergence theorem to show that
$$\int_{0}^{1}\lim_{k\to+\infty }| S_k(x)-\int_{0}^{1} f(t)\, dt|=0 $$ I find it hard, because I can not prove $lim_{k\to+\infty } S_k(x)$ converges
It seems easy but I can not figure it out. hope someone can help. Thanks in advance!