Let $X_t$ be a wide sense stationary random process indexed by $t\in\mathbb{R}$ with finite mean and variance. (http://en.wikipedia.org/wiki/Stationary_process)
Q1) Is the autocorrelation function of $X_t$ Lebesgue measurable?
Q2) Is the autocorrelation function of $X_t$ equivalent to a continuous function almost everywhere?
update:
measurability is actually equivalent to continuity a.e.