Sigma Algebras

Closure Properties and the Foundation of Measure

A \(\sigma\)-Algebra \(M\), of subsets of a set \(X\), is a collection of subsets of \(X\) that satisfies the following properties:

\begin{align} (1)&\ \text{Contains the empty set: } \emptyset \in M \\ (2)&\ \text{Closure under complementation: If } B \in M,\ \text{then } B^c \in M \\ (3)&\ \text{Closure under countable union: If } B_1, B_2, \ldots \in M,\ \text{then } \bigcup_{n=1}^{\infty} B_n \in M \end{align}

From this definition, two key facts emerge. First, since \(\emptyset \in M\), then \(\emptyset^c = X \in M\). Second, by applying De Morgan's Laws, we find that closure under countable union and complementation automatically implies closure under countable intersection:

\left(\bigcup_{n = 1}^{\infty}E_n\right)^c = \bigcap_{n = 1}^{\infty}E_n^c

Since each \(E_n^c \in M\), their intersection must also be an element of the \(\sigma\)-algebra.

Why Are \(\sigma\)-Algebras Important?

In general measure theory, an exterior measure \(m_{\ast}\) is typically only countably sub-additive. For arbitrary sets \(E_n\):

m_{\ast}\left(\bigcup_{j = 1}^{\infty}E_n\right) \leq \sum_{j = 1}^{\infty}m_{\ast}(E_n)

To perform rigorous integration, we need countable additivity (where the inequality becomes an equality for disjoint sets). We achieve this by restricting the measure to an "appropriate" \(\sigma\)-algebra.

Measurability Criterion: A set \(\mathbb{E}\) is measurable if \(\forall \epsilon > 0\), there exists an open set \(\mathbb{O} \supset \mathbb{E}\) such that:

m_{\ast}(\mathbb{O} - \mathbb{E}) \leq \epsilon

Restricting the exterior measure to subsets that satisfy this condition yields a \(\sigma\)-algebra where the measure is countably additive.

Borel \(\sigma\)-Algebra

Another vital example is the Borel \(\sigma\)-Algebra, the smallest \(\sigma\)-algebra containing all open sets of \(\mathbb{R}^d\). This includes:

\(G_{\delta}\) sets: Countable intersections of open sets.
\(F_{\sigma}\) sets: Countable unions of closed sets.

The Borel \(\sigma\)-algebra, when adjoined with null sets (sets of measure zero), forms the complete collection of Lebesgue Measurable Sets.

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