Is every analytic set $A$ in, say, $I=[0,1]$, the projection of a Borel set $B$ in, say, $[0,1] \times [0,1]$, $A = \pi_1(B)$, with the following property: For every regular Borel probability measure $\mu$ on $[0,1]$ and (EDIT) $\mu(A)=1$, there is $y \in [0,1]$ s.t.

$ \int 1_B(x,y) \mu(dx) > 0 $

where $1_B(\cdot,\cdot)$ is the indicator function of $B$?

Any results that show that analytic sets are projections of sets with sections 'large' in some sense are appreciated as well. Answers that use axioms in addition to ZFC (in particular, $V=L$) are also useful.

somesections must be small in some sense, for example if $A\subseteq [0,1]^2$ is Borel and $A_x=\{y\in [0,1]\mid (x,y)\in A\}$ is nonmeager for every $x\in [0,1]$ or if $\mu(A_x)>0$ for all $x\in[0,1]$ and some Borel probability measure $\mu$, then $\pi_1(A)$ must be Borel. In the right directionsomesection must be big in some sense, for example if $A\subseteq[0,1]^2$ is Borel and $A_x$ is $\sigma$-compact for all $x$, then $\pi_1(A)$ is Borel. $\endgroup$