What is the importance of the concept of probability space? [duplicate]

Question

We know sample space (S), elementary outcomes and events. And axiomatically define probability of these events or assign probabilities to these events. Now event space (F) and the triplet probability space (S,F,P) are defined. What is the use of probability space and event space - when we are completely done when we define the sample space and and probabilities of the events in it?

Answer 1

The first reason to work with the $\sigma$-algebra of events is in order to be able to define expectation (in fact, integration over the sample space) in a precise way. Without this notion one would have to make careful distinctions between probability spaces on which to define expectations as finite sums, series, Riemann integrals, Stieltjes integrals, etc. This becomes particularly troubling when the sample space becomes infinite dimensional, which happens all the time (think of a stochastic process $(X_i)_{i \in \mathbb N}$). The language of $\sigma$-algebras unifies and simplifies the definition of expectation.

Secondly, one can have different $\sigma$-algebras, e.g. $\mathcal F$, $\mathcal G$, defined over the same measurable space, which allows to talk about conditional expectation, $\mathbb E[X \mid \mathcal G]$, if $X$ is a random variable on $(S,\mathcal F, \mathbb P)$, which becomes interesting when $\mathcal G$ contains fewer events than $\mathcal F$. In this case $\mathbb E[X \mid \mathcal G]$ would be a function of the events in $\mathcal G$, and expresses the expected outcome of $X$ given that a particular event in $\mathcal G$ has occurred.

Answer 2

Well first I did come across this post with the same questions essentially.

Why do we need $\sigma$-algebras to define probability spaces?

So maybe that will be of help. If not, instead of asking what we gain, you might try to see if you can get around using them in the first place. For instance usually we talk about the probability of an interval say P( |X| <7) or probabilities of point say P(X=7). Like it or night we're considering some family of sets we're interested in of measuring the probability on. So this begs the question can we just ignore the fact we're using such a structure?

Unfortunately P( |X| <7) usually involves integrating over some interval e.g: (-7,7). However when talking about the probability of sets we don't want to restrict ourselves or develop methods that require us to consider events only on intervals, this would be unpractical so in comes Lebesgue integration to the rescue which is equivalent to Riemann integration over intervals.

Lebesgue integration however is based on the Lebesgue measure. For the Lebesgue measure, however the axiom of choice, which implies the existence of uncountable sets (the irrational numbers for instance cannot be made into an ordered lists), also implies that every interval contains a set that is not measurable or will violate your probability axioms if we consider that set. The standard example is called that Vital set. You can look up the construction of this set easily enough.

So to get back to your question. Most sets that people are interested in measuring the probability on are sets that are nice sets and most transformations people use are nice smooth functions so you probability won't run into too much trouble for the most part if you ignore them. However because of the generality they allow they've become intrinsic to probability theory and it's often very hard to understand the nuts and bolts of many or most statically methods without understanding this. Or in many cases how to apply these methods.

Answer 3

We P is your probability measure. S is your sample space, and F is your sigma algebra. Not all subsets of S may be measurable. However the elements of F are measurable subsets of S which are closed under unions, intersections, and complements.

Does this help?