Lebesgue Measure

Lebesgue-Stieltjes measure#

A measure on \(X\) is called Borel if its domain is the Borel \(\sigma\)-algebra on \(X\).We give a construction that yields a family of Borel measures on \(\mathbb{R}\).

Refer to sets of the form \((a,b]\) or \((a,\infty)\) or \(\empty\) where \(-\infty< a<b<\infty\) as half-open intervals (h-intervals). 

Clearly the intersection of two h-intervals is an h-interval, and the complement of an h-interval is an h-interval or the disjoint union of two h-intervals. Hence h-intervals form an elementary family on \(\mathbb{R}\). Hence the collection \(\mathcal{A}\) of finite disjoint unions of h-intervals is an algebra. It is also easy to see that the \(\sigma\)-algebra generated by \(\mathcal{A}\) is \(\mathcal{B}_\mathbb{R}\).

Proposition. Let \(F:\mathbb{R}\to\mathbb{R}\) be increasing and right continuous. If \((a_j,b_j]\) (\(j=1,…,n\)) are disjoint \(h\)-intervals, let \(\mu_0(\emptyset)=0\) and let

\[\mu_0(\cup_1^n(a_j,b_j]) = \sum_1^n[F(b_j) - F(a_j)],\]

then \(\mu_0\) is a premeasure on the algebra \(\mathcal{A}\).

From this we can first from \(F:\mathbb{R}\to\mathbb{R}\) induce a premeasure on \(\mathcal{A}\), and then by Caratheodory's theorem extend the premeasure to a Borel measure. Since \(\mathbb{R} = \cup_{-\infty}^{\infty}(j,j+1]\), the premeasure \(F\) is \(\sigma\)-finite, and hence the complete measure that is induced is unique (in the sense that there is only one Borel measure \(\mu_F\) s.t. \(\mu_F((a,b])=F(b) - F(a)\)). Clearly if \(F,G\) are increasing and right continuous and \(F-G\) is constant then \(\mu_F = \mu_G\), and the converse is also true.

The corresponding complete measure \(\overline{\mu}\), the completion of \(\mu_F\) (since \(\mu_F\) is \(\sigma\)-finite, otherwise it should be the saturation of the completion), that is obtained by restricting the outer measure \(\mu^\ast\) to the \(\sigma\)-algebra of \(\mu^\ast\)-measurable sets, is usually also denoted by \(\mu_F\), and is called the Lebesgue-Stieltjes measure associated to \(F\). When \(F(x)=x\) this is called the Lebesgue measure and we denote it by \(m\). We shall also refer to the restriction of the Lebesgues measure \(m\) to \(\mathcal{B}_\mathbb{R}\) as Lebesgue measure.