Let $f$ be a Borel function of the coordinates $x_1,\dots,x_n,\dots$ such that for all $i\in\mathbb{N}$, $$f(x_1,\ldots,x_{i-1},x_i,x_{i+1},x_{i+2},\ldots) = f(x_1,\ldots,x_{i-1},x_{i+1},x_i,x_{i+2},\ldots)$$ Let $X_1,\dots,X_n,\dots$ be independent random variables with the same distribution. We want to prove that the random variable $f(X_1,\dots,X_n,\dots)$ is a constant with probability 1.
The teacher gave me a hint that this must apparently follow from Kolmogorov's law 0 or 1, or nearby facts. But idk how to do this.