# Family of probability measures

I need a characterization of a probability space wherein the probability measure is changing.

I am not a mathematician, and do not know about stochastic processes, but I've been working through these notes: https://www.math.ku.edu/~nualart/StochasticCalculus.pdf

On page 11, there is a statement about the family of probability measures. I've inserted this as an image below. I'm stuck here; is $P_{t_1...t_n}$ one distribution, or is it a sequence of probability measures?

Sorry if this is really basic. • This is not that basic. More importantly, can you describe what application you are trying to serve by looking at a changing probability measure? – avs Dec 5 '16 at 23:10
• Those notes look handy for dusting away cobwebs, don't mind if I do :) – Mehness Dec 5 '16 at 23:29
• I don't have a concrete application, but I could invent one; I'll probably ask this as a separate question and link it here. – Learning... Dec 5 '16 at 23:30

$P_{t_1,\ldots,t_n}$ is one distribution, it is the joint distribution $$\mathbb{P} (X(t_1) \in B_1, \ldots , X(t_n) \in B_n ),$$ where $B_i$ are sets in the relevant $\sigma$-algebra ($\sigma$-field in the notes you linked). The quite remarkable thing that Kolmogorov did here is show that by specifying these probabilities for finite collections of times $(t_1,\ldots,t_n)$ we can extend this to the distribution of realisations of the whole path of the process $X(t) = X(t,\omega)$.