In Wikipedia https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula and plenty of other books/sources, Feynman-Kac formula is expressed in a form of the type $$f(t,x)=E(f(T,X_T)\mid X_t=x)$$ What is the rigorous meaning this conditional expectation conditioned on $X_T = x$ for $X_T$ with continuous distribution? Is there a definition or is it notation for something else?
In elementary courses one defines things like $E(Y\mid X=x)$ as $$ E(Y\mid X=x) = \int_{-\infty}^{+\infty}yf_{Y\mid X}(y\mid x) \, dy$$ but this definition assumes that for $(X,Y)$ there exists a joint pdf $f_{XY}$ and also that we fix a version of this pdf. It is not clear (at least to me) that changing this pdf on a set of measure $0$ will yield the same result for $E(Y\mid X=x)$.
In Karatzas-Shreve, "Brownian motion and stochastic calculus" pages 73 and 268 lies one possible answer. They prove the existence not only of the Brownian Motion but also of what they call a "Brownian family" which is the filtered space, the process, and a familily of probabilities $(P^x)_{x\in\mathbb{R^d}}$ such that for each $P^x$ the process is a Brownian motion starting at $x$ with probability $1$ (and there's also a certain measurability condition relating this probabilities) .
Then, in the case where the process $X$ is a Brownian Motion, the meaning of the expression $f(t,x)=E(f(T,X_T)∣X_t=x)$ is actually $$f(t,x)=E^{P^x}(f(T,X_{T-t}))$$ rather than "an actual conditional expectation". And the idea would be the same for other Ito processes $X$.