Let $T$ be a complete sufficient real-valued statistic for the parameter $\theta$ and $S$ be another real-valued statistic whose distribution function $F$ does not depend on the parameter $\theta$. Show that $P(S \le s\mid T)−F(s)$ equals zero almost surely under each $P_\theta$ and for each real $s$. Conclude from this that $S$ and $T$ are independent.
I'm very stuck, I don't even know where to start. I tried directly calculating $P(S \leq s\mid T)$ to get something a.s F and had no luck. I think I'm missing a theorem or property I need.
Since $T$ is sufficient for $\theta,$ the conditional distribution of $S$ given $T$ does not depend on $\theta.$ Therefore $\Pr(S\le s\mid T)$ does not depend on $\theta.$ By the law of total expectation, $\operatorname E(\Pr(S\le s\mid T)) = \Pr(S\le s).$ Therefore $\Pr(S\le s\mid T) - \Pr(S\le s)$ (since it can be computed without knowing the value of $\theta$) is an unbiased estimator of $0.$ By completeness of $T,$ this difference must therefore be $0$ with probability $1.$ And if $\Pr(S\le s\mid T) = \Pr(S\le s)$ for every value of $s,$ then $S$ and $T$ are independent.
This result is called Basu's theorem.