Probability and Stochastic Processes

1.4 Monotone Convergence Properties of Probability

Let us take a step back for a minute and comment on what we have seen thus far. -algebra differs from regular algebra in that it allows us to deal with a countable (not finite) number of sets. Again, this is a recurrent theme in probability: learning to deal with infinity. On finite spaces, things are more or less simple. One has to define the probability of each individual outcome and everything proceeds from there. However, even in these simple cases, imagine that one repeats an experiment (such as a coin toss) over and over. Again, we are forced to cope with infinity. This section introduces a way to deal with this infinity problem.

Let be a probability space.

Lemma 1.25

The following are true:

1. If and (i.e., and ), then as a ...