The space $\mathcal{M}$ of signed measures on a measurable space $(\Omega, \mathcal{F})$ is a Banach space when equipped with the total variation norm. So $(\mathcal{M},d)$ is a complete metric space, where $d$ is the total variation metric.
I thought there should be a very quick argument to show that the space of probability measures on $(\Omega, \mathcal{F})$ is of second category in the total variation topology using the Baire category theorem, but somehow I don't see how it follows. The set of probabilities isn't open, so I can't conclude immediately that it's second category. Is there some simple way to show this that I'm just missing, or does it actually require some work?
The set of probability measures is closed in $\mathcal{M}$. Indeed, it is just the set of nonnegative measures of norm $1$. The set of nonnegative measures is closed (since evaluation of a measure on any measurable set is continuous), as is the set of measures of norm $1$. So the space of probability measures is complete with respect to the metric $d$, and hence has second category in itself.