Hi, Moo!

here are a few pieces of answer...

The law of T under P is just the law of T when the probability space (here, ) is endowed with the probability measure P. In other words, this is indeed the image measure of P by T.

There are always two possible viewpoints in any probabilistic setting: either a fixed undefined very large probability space on which we define several random variables, this is the usual case in introductory courses; or a space with several distributions and the identity map as a unique random variable.- why would one be interested in the identity application ?

The interest in the first case is simplicity and universality (it doesn't matter what base space we use, so why specify); it assumes however that we rely on sometimes not obvious existence theorems.

The second case is useful when one wants to study the same random variable under various distributions. For instance, if we want to vary a parameter (like the parameter of a Bernoulli), we simply introduce a family of probabilities indexed by this parameter, and we say: "under , ...." to say that we consider the value p of the parameter. This is a very convenient setting for Markov chains, where the parameter is the starting point. We have one random variable which is the identity map on , and one measure for every site , which is the law of the Markov chain (with some given transition matrix) starting at . This allows to give meaning to expressions like (i.e. the Markov property).

This is called a Galton-Watson tree. From one ancestor (the root of the tree), we have a random number of children, each of which itself has random numbers of children, etc., where the numbers of children are all independent of each other. This is a genealogic tree where the numbers of children is random: k children (possibly 0) with probability . A basic question is: does the family eventually get extinct? i.e. is the tree finite?- how would one explain with words what the law of is ?

In this case, would be the set of trees (or a larger set), or an equivalent representation of trees. A convenient choice is to let , the set of all finite sequences of integers, and , the set of positive-integer sequences indexed by elements of . Intuitively, an individual corresponds to a sequence if it is obtained from the root as the -th child of the -th child of .... of the -th child of the root. And the number of children of this individual is encoded in the index of the tree (with arbitrary if is not connected to the root, thus there are more codings than trees).

With these notations, we can define the law of as where is the probability distribution .

Actually, what I just did is "prove" your statement, assuming that the existence of an infinite-product measure is trivial, which it is not... This is even probably the reason why this statement is outlined by the author. NB: if the author starts with such a statement, you can expect sharp rigor in the following!