I've been studying the paper "Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces" by Hunt, Sauer & Yorke, and I'm trying prove that the support of the convolution of measures (that can be assumed to be finite or probability measures) is contained in the sum of their support (footnote on page 223) without luck so far. The best my research has come up was a similar question made here: Support of the convolution of two measures, but no hint on how to prove it was given.
For the problem, let $V$ a complete metrizable TVS (not necessarily separable), consider $\mu$ and $\nu$ probability Borel measures n $V.$ The convolution of $\mu$ and $\nu$ is $$\mu * \nu(S) = \int_{V} \nu(S - x) \, d\mu(x) = \int_{V} \mu(S - y) \, d\nu(y),$$ for $S \subseteq V$ a Borel set. I want to show $supp \ \mu * \nu \subseteq supp \ \mu + supp \ \nu.$
The only thing I can think of to a approach this problem is to somehow imitate the proof for the analogous statement on the support of convolution of functions (e.g. What will be the support of the convolution of two test functions.), but it hardly seems to work without some extra assumptions on our measures. To start, the wikipedia on support of measures (also the definition of support I'm using) claims for example that $$\int_X f \, d\mu = \int_{supp\ \mu} f \, d\mu$$
if $\mu$ is a Radon measure, and the paper is not assuming inner regularity.
Any hints on how to solve this is very appreciate, thanks.