Measure

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

Some key properties of measures are:

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.

Emacs 29.4 (Org mode 9.6.15)