How Can I Learn Metropolis-Hastings?

Question

I am trying to learn Metropolis-Hastings algorithm in "Monte Carlo Statistical Methods" by Robert and Casella. I don't think i am being efficient. My teacher advised me to study chapter 6, then 7. I took undergraduate probability and measure theory, which didn't cover Radon & Nikodym, product measures, etc. My first qustion is: Is this the optimal way to understand the algorithm rigorously? I am already having problems at the start of chapter 6. I didn't understand the rigour behind the equation \begin{align*} P_x((X_1,\cdots,X_n)\in A_1\times\cdots A_n)= &\int_{A_1}\cdots\int_{A_{n-1}}K(y_{n-1},A_n)\\ &\times K(x,d(y_1))\cdots K(y_{n-2},d(y_{n-1})) \end{align*} where $K:X\times B(X)\to\mathbb{R}$ is a transition kernel and $P_x(X_1\in A)=P(X_1\in A|x)$, i.e, starting from x, probability of $X_1$ being in $A$. What i understand is that $K(x,dy)=d\mu(y)$, if we let $\mu(A):=K(x,A)$. But $K(y_1,\cdot)$ is a different measure for each $y_1\in A_1$ . So how can $K(x,d(y_1))\cdots K(y_{n-2},d(y_{n-1}))$ be a measure of $A_1\times\dots\times A_{n-1}$ ? Do i need to study measure theory a lot more to understand that? Thank you for reading.