Measure

A measure \(\mu : \Sigma \to [0,\infty]\) on a measurable space \((X, \Sigma)\) is a function satisfying:

• \(\mu(\emptyset)=0\)
• \(A_1, A_2,\cdots \in \Sigma\) such that \(A_i \cap A_j =\emptyset \implies \mu(\bigcup_i A_i) =\sum_i \mu(A_i)\) (σ-additivity/countable additivity)

Some key properties of measures are:

• \(A_1\subseteq A_2 \implies \mu(A_1)\leq \mu(A_2)\) (Monotonicity)
• \(A_1,A_2,\cdots \in \Sigma \implies \mu(\bigcup_i A_i) \leq \Sigma_i \mu(A_i)\) (Sub-additivity)
• Given \(A_1,A_2,\cdots \in \Sigma\) such that \(A_i \subseteq A_j\) for \(i
• Given \(A_1, A_2,\cdots \in \Sigma\) such that \(A_i \supseteq A_j\) for \(i

The measure \(\mu\) is called finite (or σ-finite) if \(\exists A_1,A_2,\cdots\) with \(\bigcup_i A_i = X\) such that each \(\mu(A_i)<\infty\).

There can also be non-empty sets \(Y_i\) (called null sets) such that \(\mu(Y_i)=0\) (for e.g. with the Lebesgue measure finite sets have measure 0). A property is said to hold \(\mu\) almost everywhere ($μ$-a.e.) if the property holds for all \(x \in X\textbackslash \bigcup_i Y_i\) where \(Y_i\) are nullsets.