Let $X$ and $Y$ be independent random variables and let $E$ be some subset of the real line.
I'd like to compute $P(X+Y \in E)$ using the conditional probability $P(X+c \in E|Y=c)$ and some correction term, since I have control of $P(X+c \in E|Y=c)$ via some outside information. I've always been confused by condition, disintegration etc. Does anyone have any tips?
Just use the Law of Total Probability.
If $Y$ is a discrete random variable: $$\mathsf P(X+Y\in E)=\sum_c \mathsf P(Y=c)\,\mathsf P(X+c\in E\mid Y=c)$$ Similarly if $Y$ is a continuous random variable with probability density function $f_Y(y)$, then:$$\mathsf P(X+Y\in E)=\int_\Bbb R f_Y(c)\,\mathsf P(X+c\in E\mid Y=c)\,\mathrm d c$$