What is the definition of a standard probability space?

Question

The question is basically given in the heading. For a problem, I need the existence of a measure-preserving isomorphism between the unit interval and $\textit{some}$ class of probability spaces. I encoutered the notion of a standard probability space, but could not find a textbook definition. The Wikipedia article refers to Kechris' "Classical Descriptive Set Theory", which only includes the notion of a standard Borel space. Now, it is proved that there exists a Borel isomorphism between every standard Borel space with continuous measure and the unit interval equipped with the Lebesgue measure (see theorem 17.41). My questions are: what is the difference between a standard probability and a standard Borel space, and is there $\textit{one}$ definition for a standard probability space? If yes, what is it? If no, what are the most common ones?

Answer 1

Please let me know if I understood your question correctly...

In the book Probability Theory and Examples, Durrett defines a probability space in the only way I have really ever seen it defined:

First, loosely: a probability space is a triple $(\Omega, \mathcal{F},\mathbb{P})$ where $\Omega$ is the set of outcomes, $\mathcal{F}$ is a set of events, and $\mathbb{P}:\mathcal{F} \to [0,1]$ is a function that assigns probabilities to events.

More rigorously, we stipulate that $\mathcal{F}$ is a (nonempty) $\sigma$-algebra on $\Omega$ and that $\mathbb{P}$ is a probability measure on $(\Omega,\mathcal{F})$.

In contrast, a standard Borel space must be a metrizable space that can be made complete and separable with respect to its Borel $\sigma$-algebra. Immediately, we should note that there may be sets that are measurable in our probability space that are not Borel sets. And one fact you may find useful is that if you equip any standard Borel space with a probability measure, it becomes a probability space.

However, probability spaces (in the most generality) need not be complete like standard Borel spaces. Though in most contexts, one may want to assume this.

I guess the other thing to mention is that in most applications, one assumes (in addition to the barebones definition) that your probability space satisfies the usual conditions (that is, it has what is called a filtration which satisfies a technical condition) and that your probability space can support stochastic processes like a Brownian motion. Some terms I might google would be adapted stochastic process if this bit interests you.