Hi I am studying an advanced analysis course where we have been introduced to distributions. In our case these are linear functionals on the $L_{1}^{loc}$ space.
I was wondering how these distributions are related to probability distributions or do they just share the same name?
Integration against a finite measure forms a distribution in the sense of distribution theory. And a finite measure is a probability distribution up to a normalization constant. You have probably already seen that we tend to identify a function $f$ with the distribution "integrate against $f$". We do the same thing with measures: we identify a measure $\mu$ with the distribution "integrate against $\mu$".