8  Kolmogorov: Ξ©, π’œ, β„™

8.1 🏡 The formal frame

Probability becomes rigorous when three objects are specified:

\[ (\Omega, \mathcal{A}, \mathbb{P}) \]

These symbols are the skeleton of the theory.

  • Ξ© is the sample space
  • π’œ is the collection of events
  • β„™ is the probability measure

Together, they say:

  • what may happen
  • what may meaningfully be asked
  • how likelihood is assigned

8.2 πŸ”° Ξ© β€” the space of possible outcomes

Ξ© is the universe of outcomes under the chosen model.

Examples:

  • coin toss: Ξ© = {Head, Tail}
  • die roll: Ξ© = {1,2,3,4,5,6}
  • waiting time: Ξ© = [0,\infty)
  • measurement: Ξ© = \mathbb{R}

This is the first act of modeling: define the space of possible resolution.

8.3 βš™οΈ π’œ β€” the events we are allowed to talk about

Not every subset is always manageable, especially in continuous spaces.

So Kolmogorov introduces a selected family of subsets:

\[ \mathcal{A} \subseteq \mathcal{P}(\Omega) \]

called a Οƒ-algebra.

A Οƒ-algebra is a collection of events such that:

  1. Ξ© ∈ π’œ
  2. if Event ∈ π’œ, then Event^c ∈ π’œ
  3. if Event_1, Event_2, Event_3, \dots ∈ π’œ, then

\[ \bigcup_{n=1}^{\infty} Event_n \in \mathcal{A} \]

From these rules, intersections and differences also become allowed.

⚜️ This may seem technical, but the idea is simple:

π’œ is the family of questions stable under logic and limits.

Probability needs not only possibilities, but a disciplined language of events.

8.4 ☯ β„™ β€” probability as measure

Now comes β„™.

\[ \mathbb{P}: \mathcal{A} \to [0,1] \]

This means:

  • each allowed event receives a number between 0 and 1

and that assignment must satisfy the Kolmogorov axioms:

8.4.1 1. Non-negativity

For every Event ∈ π’œ,

\[ \mathbb{P}(Event) \ge 0 \]

8.4.2 2. Normalization

The whole sample space has probability 1:

\[ \mathbb{P}(\Omega) = 1 \]

8.4.3 3. Countable additivity

If Event_1, Event_2, Event_3, \dots are pairwise disjoint, then

\[ \mathbb{P}\left( \bigcup_{n=1}^{\infty} Event_n \right) = \sum_{n=1}^{\infty} \mathbb{P}(Event_n) \]

This is the heart of the theory.

Probability respects decomposition.

8.5 πŸ’‘ Why this structure is beautiful

Kolmogorov does not tell us what reality β€œreally is.”
He does something subtler and more powerful.

He tells us what a coherent theory of uncertainty must look like.

A probability space is not a final metaphysics.
It is a formal contract between:

  • possibility
  • logic
  • measure

That is why the structure feels inevitable.

Once one has:

  • outcomes
  • events
  • stable composition of events
  • numerical assignment

the axioms almost force themselves.

8.6 πŸͺ„ Example: a fair die

Let

\[ \Omega = \{1,2,3,4,5,6\} \]

Let π’œ be all subsets of Ξ©.

Then define β„™ by:

\[ \mathbb{P}(Event) = \frac{|Event|}{6} \]

for every event Event βŠ† Ξ©.

Then:

  • β„™({2,4,6}) = 3/6 = 1/2
  • β„™({1}) = 1/6
  • β„™(\Omega) = 1

This is the finite case.

In continuous spaces the idea remains, but counting is replaced by measure.

8.7 ⚠️ Final lesson

The move from everyday chance to formal probability is the move from:

  • vague possibility
    to
  • structured possibility

The Kolmogorov framework gives probability its modern form because it joins:

  • the ontology of outcomes: Ξ©
  • the logic of events: π’œ
  • the arithmetic of uncertainty: β„™

And with that, probability becomes ready for everything that follows:

  • random variables
  • distributions
  • expectation
  • convergence
  • stochastic processes

All of them begin here.