Let be $(X, \mathcal{A}, \mu)$ a measure space and $f: X \to [0, +\infty]$ a measurable function such that for every $x \in X$ we have $f(t + x) = f(t)$ for almost every $t \in X$. Is true that $f$ is constant? or constant almost everywhere?
Thanks in advance.
Hints: as a simple consequence of Fubini's Theorem, for almost all $t$, the equation $f(x+t)=f(t)$ holds for almost all $x$. [Use the fact that $\int \int |f(x+t)-f(t)| dx dt =\int \int |f(x+t)-f(t)| dt dx$ ]. In particular this holds for one value of $t$ and hence $f$ is constant almost everywhere. Can you see that $f$ need not be constant? [Take characteristic function of a non-empty set of measure $0$]. Will be happy to provide a detailed answer if needed.