Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\sigma$-algebra, and let $\mathbb P$ be a probability measure on $(\Omega, \mathcal F)$.

Denote by $\vec x = (x,y)$ a point in $\Omega$, and let $\mathcal F_1$ be the $\sigma$-algebra generated by the first coordinate. Fix $y_0 \in \mathbb R$ and $\eta > 0$, and consider $$f(\vec x) = \mathbb P( ~|y - y_0| \le \eta~ |\mathcal F_1).$$ (More generally, one can consider $f(\vec x) = \mathbb E( \varphi(\vec x) | \mathcal F_1)$ for some suitable $\varphi : {\mathbb R}^2 \to \mathbb R$.)

The function $f$ is measurable; that comes from the definition of conditional expectations. I would like to find some reasonable sufficient conditions such that $f$ is continuous and positive.

I feel like this should be relatively elementary material, but unfortunately I'm having trouble finding any references. How should I approach this?

_{I've included the [fa.functional-analysis] tag because in general I want to consider $\Omega$ to be a space of smooth functions. I'm guessing that to give some additional structure, I'll need to assume that $\mathbb P$ is absolutely continuous with respect to a Gaussian measure, because I don't know any other reasonable measures on function space.}