Quickest Way For Me To Learn About Metropolis Hastings First of all, thanks for reading. I have a month to learn about Metropolis-Hastings with mathematical rigour, and i don't have other responsibilities. I am using second edition of "Monte Carlo Statistical Methods" by Robert and Casella that my professor recommended me. My teacher advised me to study chapter 6 (Markov Chains), then 7 (Metropolis-Hastings). I took undergraduate probability, which covered markov chains with transition matrices (discrete state space), 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 transition kernels. Particularly 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 we integrate $K(x,d(y_1))\cdots K(y_{n-2},d(y_{n-1}))$ over $A_1\times\dots\times A_{n-1}$ ? Do i need to study measure theory a lot more to understand that?
 A: 
My first qustion is: Is this the optimal way to understand the algorithm rigorously?

It is not possible for any of us to explain the equations without seeing all the previous context, notational definitions, etc.  Regardless of this, I'm not a subscriber to the idea that you have to drench relatively simple ideas with a firehose of measure theory notation before they are "understood rigorously".  My experience is that this often makes statistical methods less well understood.  If you are having trouble with the measure theoretic presentation, see if you can go through the material and translate it back into regular statistical notation without the use of measures (i.e., using density functions, distribution functions, etc.).  Do this even if it entails a loss of generality, since you can always start from a less general presentation that makes sense to you, and then generalise from there using measures.  You might find that the Lebesgue-Stieltjes integral is a good halfway method for this.
Another thing to bear in mind when learning algorithms is that it is far less important to understand the measure-theoretic details than it is to be able to actually construct and use the algorithm (and have some intuitive idea of why it works).  I'm a big fan of learning by doing, especially when dealing with algorithms.  My recommendation would be to set aside the measure theory work for a while and muddle through an attempt to program the Metropolis-Hastings algorithm for a simple problem where you can compare it to a known solution (i.e., a problem where the algebra yields a closed form integral).  Once you have done this you will be able to see why it is working and this will assist your later mathematical analysis.  Seperately from that, it will also give you more holistic applied knowledge of the method.
