Is regular conditional distribution unique?

Question

If $\kappa_1,\kappa_2:\mathfrak{B}\times\Omega\rightarrow\left[0,1\right]$ are two regular versions of the same conditional distribution $P\left(\left.X\in\cdot\right|\mathcal{A}\right)$, are they equal as functions?

Answer 1

If I understood your question correctly, then the answer is "no". Let $\mu$ be a Lebesgue measure on $[0,1]$ with Borel $\sigma$-algebra and consider two functions $f(x,y) = x$ and $$ g(x,y) = 1(x\neq 1)f(x,y). $$ Both functions imply the same joint distribution, but they are not equal.