Generated σ-algebras: what it means (and common my confusions)
This note is about a specific phrase that shows up everywhere in measure theory/probability: “generated”.
Basic definitions: σ-algebras, π/λ-systems, and the monotone class theorem.
What “generated” always means
Given a set $\Omega$ and a collection $\mathcal A \subseteq \mathcal P(\Omega)$, the σ-algebra generated by $\mathcal A$ is
$$ \sigma(\mathcal A) := \bigcap\{\mathcal G \subseteq \mathcal P(\Omega) : \mathcal G \text{ is a σ-algebra and } \mathcal A \subseteq \mathcal G\}. $$Interpretation: $\sigma(\mathcal A)$ is the smallest σ-algebra containing $\mathcal A$.
“$\mathcal F$ is generated by $\mathcal F_0$” (the precise meaning)
Let $(\Omega,\mathcal F)$ be a measurable space and let $\mathcal F_0 \subseteq \mathcal P(\Omega)$ be any family of subsets. Then:
$$ \text{“$\mathcal F$ is generated by $\mathcal F_0$”} \quad \Longleftrightarrow \quad \mathcal F = \sigma(\mathcal F_0). $$Once $\mathcal F = \sigma(\mathcal F_0)$ holds, it automatically implies $\mathcal F_0 \subseteq \mathcal F$ (because $\mathcal F_0 \subseteq \sigma(\mathcal F_0)$).
Common confusion: “$\mathcal F_0$ is a collection of subsets of $\Omega$, so it must be inside $\mathcal F$”
Not necessarily.
- $\mathcal F \subseteq \mathcal P(\Omega)$ is the collection of sets we decide to call measurable.
- An arbitrary $\mathcal F_0 \subseteq \mathcal P(\Omega)$ might contain sets that are not in $\mathcal F$ (non-measurable w.r.t. that $\mathcal F$).
What is true is:
- If we assume $\mathcal F_0 \subseteq \mathcal F$, then $\sigma(\mathcal F_0)\subseteq \mathcal F$ because $\mathcal F$ is itself a σ-algebra containing $\mathcal F_0$.
So “$\mathcal F_0 \subseteq \mathcal F$” gives “$\sigma(\mathcal F_0)\subseteq\mathcal F$”, but not necessarily equality.
Common confusion: “if $\mathcal F_0$ generates $\mathcal F$, does $\mathcal F_0$ list every subset of $\Omega$?”
No.
$\mathcal F_0$ generating $\mathcal F$ means:
- every measurable set $A\in\mathcal F$ can be built from $\mathcal F_0$ using complements and countable unions (and thus countable intersections).
It does not mean:
- every subset of $\Omega$ is measurable; i.e. it does not imply $\mathcal F = \mathcal P(\Omega)$.
The power set $\mathcal P(\Omega)$ is one possible σ-algebra, but in most interesting measure/probability settings $\mathcal F\neq \mathcal P(\Omega)$.
Common confusion: “σ-algebras can only handle countable sets”
Be careful about what “countable” refers to.
- A σ-algebra can contain uncountable sets (intervals, Cantor set, etc.).
- The axiom is closure under countable unions (i.e. union of countably many sets), not “closure only for countable sets”.
Two standard examples to keep in mind
Example 1: Borel σ-algebra on $\mathbb R$
Let
$$ \mathcal F_0 := \{(a,b) : a \lt b\} $$be the collection of open intervals. Then
$$ \sigma(\mathcal F_0)=\mathcal B(\mathbb R), $$the Borel σ-algebra.
Even though $\mathcal F_0$ is “small”, $\mathcal B(\mathbb R)$ contains many sets not explicitly listed in $\mathcal F_0$. For instance, every singleton $\{x\}$ is Borel since
$$ \{x\}=\bigcap_{n\ge 1}\left(x-\frac1n,\,x+\frac1n\right), $$a countable intersection of open intervals.
Example 2: σ-algebra generated by a random variable
If $X:\Omega\to\mathbb R$ is a random variable (measurable map), then the σ-algebra generated by $X$ is
$$ \sigma(X):=\{X^{-1}(B):B\in\mathcal B(\mathbb R)\}. $$Interpretation: $\sigma(X)$ is exactly the family of events whose truth value can be decided if we only know the value of $X$.
A precise “minimal generator” statement (one possible meaning)
People say “$\mathcal F_0$ is minimal” in different ways; one common meaning is minimal by inclusion:
$$ \sigma(\mathcal F_0)=\mathcal F \quad\text{and}\quad \forall A\in\mathcal F_0,\; \sigma(\mathcal F_0\setminus\{A\})\neq \mathcal F. $$Even under this “minimal” condition, we still only generate $\mathcal F$, not $\mathcal P(\Omega)$ (unless $\mathcal F=\mathcal P(\Omega)$ in the first place).