It is well known that meagre sets (in topology theory) and sets of measure zero (in measure theory) are generally not the same things ([O], see also a related question on MO). A set that is small in one sense may be large in the other sense. But surely, from the names suggested, should the opposite be true at least for some common regular measures? In particular, let $X$ be a locally convex metrizable space, I would like to know

(i) If $A$ is a set of the first category of $X$, does there exist a (probability) regular measure $\mu$ on $X$ such that $\mu(A) = 0$?

(ii) Would the Gaussian measure on $X$ be a positive answer for (i)?

**Reference:**
[O] *Oxtoby, J. C.*, Measure and category. A survey of the analogies between topological and measure spaces, Graduate Texts in Mathematics. 2. New York-Heidelberg-Berlin: Springer- Verlag. VIII, 95 p. (1971). ZBL0217.09201.

everymeager set is null? $\endgroup$no: mathoverflow.net/questions/342798/… $\endgroup$4more comments