Foundations — Probability space — sample space Ω, sigma-algebra F, measure P — Kolmogorov axioms

Before you can read the parent note Probability space, you must recognise every symbol it throws at you. This page builds them from zero, in an order where each one leans only on the ones before it.

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0. The alphabet of "sets" (the true starting point)

Everything in probability is built from sets — collections of things. So we start there.

Look at figure s01: the lavender blob is a set, each dot is an element. The dot with the arrow satisfies $x\in S$; the dot outside satisfies $y\notin S$. That is all the symbol $\in$ ever means.

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1. Combining sets: complement, union, intersection

These three operations are the entire grammar of "events". The parent uses all of them constantly.

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2. Infinite lists and the symbols $\bigcup$, $\bigcap$, $\sum$

The parent doesn't stop at two events — it uses infinitely many. Three "big operator" symbols handle that.

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3. The three headline symbols: $\Omega$, $\mathcal{F}$, $P$

Now the actual objects of the parent note. Each is built from Sections 0–2.

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4. The measurement symbols you'll meet next door

The parent's examples lean on two more notations. Meet them now.

Recall Check: is

$\mathcal{F}$ ever smaller than $2^{\Omega }$? Yes ::: On uncountable $\Omega$ (like $[0,1]$) non-measurable sets force $\mathcal{F}$ (e.g. the Borel sigma-algebra) to be a strict subset of $2^{\Omega }$.

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5. How the foundations feed the topic

The map reads bottom-up: raw set language builds $\Omega$, $\mathcal{F}$, and $P$; bundling them gives the triple; the axioms live on top; and from there the rest of probability — Conditional probability & independence, Random variables, Expectation and Lebesgue integral — grows. The Inclusion–exclusion principle and Measure theory & Lebesgue integration extend the same machinery.

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Equipment checklist

Cover the right-hand side and test yourself. If any is fuzzy, re-read its section above before opening the parent.

• $x\in S$ means ::: $x$ is an element of (is inside) the set $S$.
• $A\subseteq B$ means ::: every element of $A$ is also in $B$; $A$ sits entirely inside $B$.
• $\varnothing$ is ::: the empty set — no elements — modelling the impossible event.
• $A^c$ is ::: the complement of $A$: everything in $\Omega$ that is not in $A$ ("not $A$").
• $A\cup B$ is ::: the union: outcomes in $A$ or $B$ or both ("at least one").
• $A\cap B$ is ::: the intersection: outcomes in both $A$ and $B$ ("both").
• "$A$ and $B$ are disjoint" means ::: $A\cap B=\varnothing$; they share no outcome, so probabilities may be added.
• ${\bigcup }_{i=1}^{\infty }{A}_i$ is ::: outcomes lying in at least one $A_i$ of an infinite (countable) list.
• ${\bigcap }_{i=1}^{\infty }{A}_i$ is ::: outcomes lying in all of the $A_i$ at once.
• ${\sum }_{i=1}^{\infty }{p}_i$ is ::: the running total $p_1+p_2+p_3+\cdots$ of the numbers.
• "countable" means ::: labellable one-by-one by $1,2,3,\dots$, even if the list never ends.
• $\Omega$ is ::: the sample space — the set of all possible outcomes $\omega$.
• $\mathcal{F}$ is ::: the sigma-algebra — the collection of subsets of $\Omega$ we call events.
• $P:\mathcal{F}\to [0,1]$ means ::: $P$ takes an event and returns its probability, a number from 0 to 1 inclusive.
• $(\Omega ,\mathcal{F},P)$ is ::: the probability space — the three ingredients bundled as one package.
• $|A|$ is ::: the number of elements in a finite set $A$.
• $2^{\Omega }$ is ::: the power set — all subsets of $\Omega$; has $2^n$ members if $|\Omega |=n$.