Set-up
Let $X$ and $Y$ be independent random variables having densities $f_X$ and $f_Y$, which also have a joint density, with respect to the Lebesgue measure.
Question
I am trying to derive a formula for the density $X$ conditional on $X+Y$ following the argument in Conditional density of Sum of two independent and continuous random variables
Questions like this or this consider the conditional expectation given the sum.
I am trying to derive the density and I am very confused.
Calculations
We want to find a function $f_{X|X+Y}$ such that $$ E[1_{\{X\in A\}}|\sigma(X+Y)](\omega) \stackrel{P-\text{a.s.}}{=} E[1_{\{X\in A\}}|X+Y=t]|_{t=X(\omega)+Y(\omega)}=\int_{\mathbb{R}}1_{A}(x)\,f_{X|X(\omega)+Y(\omega)}(x)dx \quad\text{for any }A\in\mathcal{B}(\mathbb{R}). $$ From the factorization lemma, there exists a measurable function $\varphi_A\colon\mathbb{R}\to \overline{\mathbb{R}}$ such that $$ E[1_{\{X\in A\}}|X+Y=t]=\varphi_A(t). $$ Now, $B\in\mathcal{B}(\mathbb{R})$ implies $\{X+Y\in B\}\in \mathcal{B}(\mathbb{R})$, and thus from the definition of the conditional expectation we have \begin{align} \int_{\mathbb{R}}\!\int_{\mathbb{R}} 1_A(x)1_{B}(x+y)f_X(x)f_Y (y)\,dxdy&\\= E[1_{\{X\in A\}}\,1_{\{X+Y\in B\}}] &=E[(E[1_{\{X\in A\}}|\sigma(X+Y)])1_{\{X+Y\in B\}}]\\ &=E[\varphi_A(X+Y)1_{\{X+Y\in B\}}]\\ &=\int_{\mathbb{R}}\varphi_A(t)1_{B}(t)f_{X+Y}(t)dt \\ %&\stackrel{\text{also we have}}{=}E\Bigg[\Bigg(\int_{\mathbb{R}} 1_{A}(x)\,f_{X|X(\omega)+Y(\omega)}(x)dx\Bigg)\,1_{\{X+Y\in B\}}\Bigg]\\ %&=\int_{\mathbb{R}}\!\int_{\mathbb{R}}\Bigg(\int_{\mathbb{R}} % 1_{A}(x)\,f_{X|x'+y}(x)dx\Bigg)1_{B}(x'+y)f_X(x')f_Y(y)\,dx'dy \end{align} From here the aforementioned answer would obtain a form of $\varphi_A$ by an identification. But here $f_{X+Y}$ is the convolution of $f_X$ and $f_Y$, and I do not know how to proceed...