I don't know where to start with this problem:
Let $(c_n)_{n\in\mathbb{N}}$ be a sequence that satisfies $\sum c_n^2<\infty$ and $r_n(t)$ be the Rademacher function (i.i.d. random +1, -1 signs).
Define: $$f_t(x) =\sum_{n=0}^\infty r_n(t)c_n\cos(nx) $$ Show that $f_t(x)\in L^p([0,2\pi])$ for almost all $t\in[0,1]$.
Any ideas?
Ok, if the OP hasn't got it yet he's not trying. For everyone else: Khinchine says that $$\int_0^1|\sum r_n(t)\alpha_n|^p\le c_p\left(\sum|\alpha_n|^2\right)^{p/2}$$ for any $\alpha_n\in\mathbb C$. So Fubini, or more properly Tonelli, shows that $$\begin{aligned}\int_0^1\int_0^{2\pi}|f_t(x)|^p\,dxdt&=\int_0^{2\pi}\int_0^{1}|f_t(x)|^p\,dtdx\\&\le c_p\int_0^{2\pi}\left(\sum|c_n\cos(nx)|^2\right)^{p/2}\,dx \\&\le c_p\int_0^{2\pi}\left(\sum|c_n|^2\right)^{p/2}\,dx \\&=2\pi c_p\left(\sum|c_n|^2\right)^{p/2} \\&<\infty.\end{aligned}$$
Hence $\int_0^{2\pi}|f_t(x)|^p<\infty$ for almost every $t$.
Curiously, not really about Fourier series at all; orthogonality of those cosines has nothing to do with it, the same thing works with any uniformly bounded sequence of functions in place of $\cos(nx)$.