I am taking a course on Stochastic Analysis, and the introduction of the course covers Measure Theory\ Probability Theory. I am very new to this area of abstract maths and am trying to get to grips with the abstract approach to mathematics. I've been provided with the following definition regarding completeness of a probability space:
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $A \subset \Omega$ (not necessarily an element of $\mathcal{F}$). We say that $A$ is a null set if there exists $B \in \mathcal{F}$ such that $A \subset B$ and $\mathbb{P}(B)=0$. We denote by $\mathcal{F}^\mathbb{P}$ the family of all subsets of $\Omega$ of the form $A \cup C$ where $A \in \mathcal{F}$ and $C$ is a null set. The collection $\mathcal{F}^\mathbb{P}$ is called the completion of $\mathcal{F}$ with respect to $\mathbb{P}$. The probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is called complete if $\mathcal{F} = \mathcal{F}^\mathbb{P}$.
Firstly, I am struggling to get to grips with this definition, as it is so heavy on notation/ maths, but moreover, what is the relevance of completion of a sigma algebra or a probability space? If anyone can break down this definition into more simplistic/ plain wording, and provide some insight into the reason for dealing with so called 'completeness', that would be brilliant.
The intuition is that sets $B \in \mathcal{F}$ such that $\mathbb{P}(B)=0$ are irrelevant. Following this intuition, we would like that if $A \subset B$ and $B$ is irrelevant then $A$ is irrelevant. However, there is a "small problem": $A$ may not be in $\mathcal{F}$ and so $\mathbb{P}(A)$ may not be defined.
Of course, we could try to simply define that any subset of an irrelevant set is also irrelevant, but then we will need to careful in several moments.
For instance, suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is not complete. Let $X: \Omega \rightarrow \Bbb R$ be a random variable (a measurable function) and $Y: \Omega \rightarrow \Bbb R$ be a function. Suppose that there is $B \in \mathcal{F}$ such that $\mathbb{P}(B)=0$ and $[X\neq Y] \subset B$ (where $[X\neq Y]= \{x \in \Omega: X(x) \ne Y(x) \}$) . It means $X$ and $Y$ differs only in a subset of an irrelevant set. We say that $X=Y$ almost surely. However, $(\Omega, \mathcal{F}, \mathbb{P})$ is not complete, $[X\neq Y]$ may not in $\mathcal{F}$ and $Y$ may not be measurable, so $Y$ may not be a random variable.
How could we really make that subsets of irrelevant sets to be irrelevant? Answer: by adding them to $\mathcal{F}$ (which is $\mathcal{F}^\mathbb{P}$) and then extending $\mathbb{P}$ to those subsets by assigning them value 0. This is called the completion of $(\Omega, \mathcal{F}, \mathbb{P})$. A measure space is complete is it is equal to its completion.
If $(\Omega, \mathcal{F}, \mathbb{P})$ is complete, $X: \Omega \rightarrow \Bbb R$ is a random variable (a measurable function), $Y: \Omega \rightarrow \Bbb R$ is a function and $X=Y$ almost surely, then $Y$ is a random variable.