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Let

  • $(\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$
  • $X$ be real-valued random variable on $(\Omega,\mathcal A,\operatorname P)$
  • $\kappa$ be a Markov kernel on $(\mathbb R,\mathcal B(\mathbb R))$
  • $p\in[0,1]$

Assume we construct a real-valued random variable $Y$ on $(\Omega,\mathcal A,\operatorname P)$ in the following way: With probability $p$ we draw $Y$ from $\mu$ and with probability $1-p$ we draw $Y$ from $\kappa(X,\;\cdot\;)$.

What's the conditional distributon $\operatorname P\left[Y\in\;\cdot\;\mid X\right]$ of $Y$ given $X$? In particular, I want to determine the Markov kernel $Q$ on $(\mathbb R,\mathcal B(\mathbb R))$ such that $$\operatorname P\left[Y\in B\mid X\right]=Q(X,B)\;\;\;\text{almost surely for all }B\in\mathcal B(\mathbb R).\tag1$$

In order to give a rigorous answer, I think that we need to introduce a $\{0,1\}$-valued $p$-Bernoulli distributed random variable $Z$ on $(\Omega,\mathcal A,\operatorname P)$ such that

  1. $X$ and $Z$ are independent
  2. $X$ and $Y$ are independent given $\{Z=1\}$
  3. $\operatorname P\left[Y\in B\mid Z=1\right]=\mu(B)$ for all $B\in\mathcal B(\mathbb R)$
  4. $\operatorname P\left[Y\in B\mid X\right]=\kappa(X,B)$ almost surely on $\{Z=0\}$ for all $B\in\mathcal B(\mathbb R)$

At first glance, I thought this would be an easy task. However, I don't know how I need to proceed. First of all, is my (supposed to be equivalent) description of the problem with the random variable $Z$ correct or did I impose any false assumption?

If the description is correct, how do we need to proceed?

Please take note of this related question: https://math.stackexchange.com/q/3305603/47771.

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