How to get rid of the expectation in Monte Carlo Policy Gradient method?
But how he removed the expectation part when talking stochastic gradient decent ?


A result that shows what happens in expectation can be estimated by sampling it, and that is precisely what stochastic gradient descent (or ascent in this case) methods do - they operate on individual samples on the assumption that this will on average produce reasonable direction for optimising parameters. 

So there is no need to "get rid of" the expectation. In fact the sequence of equations is working deliberately towards it, because in most situations we do not know the full characteristics of the MDP and cannot calculate $d(s)$ (and maybe not even $\mathcal{R}_{s,a}$) - the expectation is used to remove the need for knowing those terms.

Importantly, $d(s)$ is the probability density of the state = the likelihood on a random sample of all states visited under a certain policy of finding the agent/environment in that state. This is often hard to calculate directly, even if you know the MDP and the policy, and it is intractable if you do not have a model of the MDP. However, when you take many samples then just by the act of sampling and using the value of $s$ that is observed, then you will approximate the true distribution of states in the long run.