Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. Consider the following SDE $$dX_t=\mu(X_t)dt+\sigma (X_t)dB_t.\tag{E}$$
Suppose that $\mu$ and $\sigma $ are nice enough so that it has a solution (for instance Lipschitz with linear growth).
We denote $(X_t^x)$ the solution of $(E)$ that start at $x$ almost surely.
In this case, in my lecture, we define $\mathbb P^x$ being the measure defined by $$\mathbb P^x(X_t^0\in A)=\mathbb P(X_t^x\in A).$$
Why do we do that ? In what the measure $\mathbb P^x$ are interesting ? First, I was thinking that it helps to simplify notation, but not really at the end, because to write $\mathbb P^x(X_t^0\in A)$ or $\mathbb P(X_t^x\in A)$ is really the same (it's even quicker to write $\mathbb P(X_t^0\in A)$). So, why these measures $\mathbb P^x$ ? (because these $\mathbb P^x$ are very often used).
One main reason is Markov property. It is less cumbersome/more-readable to write
$$E_{x}[f(X_{T})|X_{t}]=E_{X_{t}}[f(X_{T-t})]$$
than
$$E_{x}[f(X_{T})|X_{t}]=E[f(X_{T-t}^{X_{t}})].$$