Measurable space

If $\mathrm{\Omega }$ is a given set, then a $\sigma$-algebra $\mathcal{F}$ on $\mathrm{\Omega }$ is a family $\mathcal{F}$ of subsets (called measurable sets) of $\mathrm{\Omega }$ with the following properties:

1. Empty Set:
   $\mathrm{\varnothing }\in \mathcal{F}$
2. Closed under Complements:
   $F\in \mathcal{F}\Rightarrow F^C\in \mathcal{F}$, where $F^C=\mathrm{\Omega }\setminus F$ is the complement of $F$ in $\mathrm{\Omega }$
3. Closed under Countable Unions:
   $A_1,A_2,\dots \in \mathcal{F}\Rightarrow A:=\bigcup _{i=1}^{\mathrm{\infty }}{A}_i\in \mathcal{F}$

The pair $(\mathrm{\Omega },\mathcal{F})$ is called a measurable space.

Here we can define a measure and obtain a measure space. measurable function

Interpretation

A $\sigma$-algebra $\mathcal{F}$ is like a "system of holes" or a "viewing window" that you get to look through to see the outcome of a experiment.
Suppose we roll a the die ($\mathrm{\Omega }=\{1,2,3,4,5,6\}$):

• The "Experiment" ($\mathrm{\Omega }$): This is a dark room where the die has been rolled. The outcome (e.g., a "4") is in the room, but you're outside.

• The "Wall": This is a barrier for information between you and the outcome.

• The $\sigma$-Algebra ($\mathcal{F}$): This is the set of holes you're allowed to look through.

Scenario 1: The "Full Info" Wall (${\mathcal{F}}_{\text{full}}$)

• Holes: The wall is made of clear glass. You can see the die perfectly.

• What you can observe: You can answer any question. "Was it a 4?" (Yes). "Was it even?" (Yes). "Was it less than 3?" (No).

• This $\sigma$-algebra is the "power set"—it contains all possible subsets as "viewable" events. Your observation is perfect.

Scenario 2: The "Even/Odd" Wall (${\mathcal{F}}_{\text{even}}$)

• Holes: The wall is solid, but it has two peepholes.

  • One is labeled "ODD" ($\{1,3,5\}$).

  • One is labeled "EVEN" ($\{2,4,6\}$).

• What you can observe: A light flashes in the room. You can't see the die itself, but you can see which peephole the light is coming from.

• Measurable Events ("Questions you can answer"):

  • "Was the roll even?" Yes, you can answer this. If you see the light in the "EVEN" hole, the answer is yes. This event ($\{2,4,6\}$) is measurable with this $\sigma$-algebra.

  • "Was the roll a 4?" No, you cannot answer this. The light in the "EVEN" hole tells you the outcome was in the set $\{2,4,6\}$, but you can't distinguish which of those three it was. The event $\{4\}$ is not measurable with this $\sigma$-algebra.

Scenario 3: The "No Info" Wall (${\mathcal{F}}_{\text{none}}$)

• Holes: The wall is solid. There are no holes.

• What you can observe: You know a die was rolled (this is $\mathrm{\Omega }$, the whole room), but you see nothing.

• Measurable Events: The only question you can answer is "Did something happen?" (Yes, $\mathrm{\Omega }$). This is the trivial $\sigma$-algebra $\{\mathrm{\varnothing },\mathrm{\Omega }\}$.