Suppose we have random variables $U, V, Y$, where $Y$ is real-valued, and some event $A$ such that, almost surely, $$ U \, \mathbb{1}_A = V \, \mathbb{1}_A $$ where $\mathbb{1}_A$ is an indicator function. Does it follow that $$ \mathbb{E}[Y \mid \mathbb{1}_A = 1, U] \, \mathbb{P}(A \mid U) = \mathbb{E}[Y \mid \mathbb{1}_A = 1, V] \, \mathbb{P}(A \mid U) $$ almost surely?
If everything is discrete, then, provided $\mathbb{P}(U = u) > 0$, we have $$ \mathbb{E}[Y \mid \mathbb{1}_A = 1, U = u] \, \mathbb{P}(\mathbb{1}_A = 1 \mid U = u) %= \left(\sum_{y} y \, \mathbb{P}(Y = y \mid \mathbb{1}_A=1, U=u)\right)\mathbb{P}(\mathbb{1}_A = 1 \mid U = u) \\ = \frac{\sum_y y \, \mathbb{P}(Y = y, \mathbb{1}_A=1, U=u)}{\mathbb{P}(\mathbb{1}_A=1, U=u)} \, \mathbb{P}(\mathbb{1}_A = 1 \mid U = u)\\ = \frac{\sum_y y \, \mathbb{P}(Y = y, \mathbb{1}_A=1, V=u)}{\mathbb{P}(\mathbb{1}_A=1, V=u)} \, \mathbb{P}(\mathbb{1}_A = 1 \mid U = u) \\ = \mathbb{E}[Y \mid A = a, V = u] \, \mathbb{P}(\mathbb{1}_A = 1 \mid U = u). $$ I am wondering if the same holds true in general, i.e. under the general definition of conditional expectation, with $Y$ an arbitrary (integrable) random variable, and $U$ and $V$ arbitrary measurable functions (subject to the condition above).
You can say that $$ \mathsf{E}[Y\mid 1_A,U]=\mathsf{E}[Y\mid 1_A,V] \quad\text{a.s. on }A $$ because for any bounded, measurable function $g$, \begin{align} &\mathsf{E}[1_A \mathsf{E}[Y\mid 1_A,U]g(1_A,V)] \\ &\qquad=\mathsf{E}[\mathsf{E}[1_A Y\mid 1_A,U]g(1_A,U)] \\ &\qquad=\mathsf{E}[1_AY g(1_A,U)] \\ &\qquad=\mathsf{E}[1_AY g(1_A,V)] \\ &\qquad=\mathsf{E}[\mathsf{E}[1_A Y\mid 1_A,V]g(1_A,V)] \\ &\qquad=\mathsf{E}[1_A \mathsf{E}[Y\mid 1_A,V]g(1_A,U)]. \end{align}