1.5Borel Measures of the Real Line

Updated 28 Sep 2026Chapter (PDF)

[1.5.1]Definition(Borel Measure)#

A Borel measure is a measure whose domain is a Borel $\sigma$ -algebra for the topological space $(X, \Tau)$ .

As such, a Borel measure on $\bR$ is simply a measure whose domain is the Borel $\sigma$ -algebra generated by the standard topology on $\bR$ . We will motivate the section by first introducing what is known as a cumulative distribution function. If $\mu$ is a finite Borel measure on $\bR$ , then we may define the cumulative distribution $F(x) = \mu((-\infty, x])$ . $F(x)$ accumulates the value of the measure $\mu$ at all points along the real line, similar to how a cumulative distribution function in probability accumulates the values of a probability measure across an event space. Notably, $F$ starts from $0$ , is increasing, and is right continuous as follows.

If $x < y$ , then $(-\infty, x] \subseteq (-\infty, y]$ , which implies $F(x) = \mu((-\infty, x]) \le \mu((-\infty, y]) = F(y)$ , and so $F$ is monotonically increasing. Moreover, $F$ is right continuous. Indeed, if $x_n \searrow x$ is a monotonically decreasing sequence converging to $x$ , then note that the intervals $(-\infty, x_n]$ form a decreasing sequence of measurable sets whose intersection is exactly $(-\infty, x]$ . That is, $\bigcap_{n=1}^\infty (-\infty, x_n] = (-\infty, x]$ . Because $\mu$ is a finite measure, we can invoke continuity from above to conclude that

\lim_{n \to \infty} F(x_n) = \lim_{n \to \infty} \mu((-\infty, x_n]) = \mu\left(\bigcap_{n=1}^\infty (-\infty, x_n]\right) = \mu((-\infty, x]) = F(x).

Since $\lim_{n \to \infty} F(x_n) = F(x)$ holds for any sequence $x_n \searrow x$ , it follows that $F$ is right continuous.

Moreover, if $b > a$ , then note $(-\infty, b] = (-\infty, a] \cup (a, b]$ , and by countable additivity, we have that

\mu((-\infty, b]) = \mu((-\infty, a]) + \mu((a, b]),

and so $\mu((a, b]) = F(b) - F(a)$ .

Noting all of this, we work backwards. Given a function $F: \bR \to \bR$ that is increasing and right-continuous, we seek a Borel measure on $\bR$ generated by $F$ , say $\mu_F$ , satisfying $\mu_F((a, b]) = F(b) - F(a)$ as from before. This is called the Lebesgue-Stieltjes construction. Eventually, we will show this construction works in reverse as well—that is, for an arbitrary Borel measure $\mu$ , there is some function $F: \bR \to \bR$ such that $\mu = \mu_F$ . This is called the Lebesgue-Stieltjes correspondence.

The correspondence isn't necessarily unique, however; indeed, two functions may generate the same Borel measure under the Lebesgue-Stieltjes construction, and accordingly, an arbitrary Borel measure may give rise to multiple functions that induce it. Of course, this would be problematic if the various functions generating a Borel measure were completely different (and vice versa), as there isn't even much of a correspondence if there is no determinism. Thankfully, the “various functions” differ by only an additive constant, as we will state more formally later.

In any case, the Lebesgue-Stieltjes construction and correspondence will lead to a rich theory of assigning length to Borel sets in $\bR$ . As a specific case, if our function is given by $F(x) = x$ , the measure we derive is the Lebesgue measure, the standard way to assign length to Borel sets of $\bR$ .

We develop the theory of the Lebesgue-Stieltjes construction by considering the half-open intervals $(a, b]$ and rays $(a, \infty)$ and $(-\infty, a)$ . More specifically, we consider the collection of all half-open intervals and rays, and close this set under complementation and finite unions. A general element of this algebra is then simply a finite disjoint union of half-open intervals and rays. Given a function $F: \bR \to \bR$ , we define a premeasure on this algebra that extends to a measure whose distribution function is $F$ , up to an additive constant.

[1.5.2]Theorem(Increasing Right-Continuous Function Induces a Premeasure)#

Let $F: \bR \to \bR$ be increasing and right-continuous. Let $\mathcal A$ be the algebra of finite disjoint unions of half-open intervals and rays. For a standard half-open interval $I = (a, b]$ , define

\mu_0(I) = F(b) - F(a).

The measure of a ray follows inevitably from the countable additivity required of any premeasure on disjoint sets whose union remains in the algebra. By partitioning a ray into a countable union of half-open intervals—for instance, $(a, \infty) = \bigsqcup_{k=1}^\infty (x_{k-1}, x_k]$ where $a = x_0 < x_1 < x_2 < \dots$ and $x_n \to \infty$ —the measure evaluates as a telescoping sum:

\mu_0((a, \infty)) = \sum_{k=1}^\infty \left[ F(x_k) - F(x_{k-1}) \right] = F(\infty) - F(a).

By symmetry, $\mu_0((-\infty, b]) = F(b) - F(-\infty)$ . For the empty set, $\mu_0(\emptyset) = 0$ .

Let $A \in \mathcal{A}$ be arbitrary, and write $A = \bigcup_{k = 1}^n A_k$ , with $\{A_k\}_{k = 1}^n$ a collection of pairwise disjoint half-open intervals or rays. Define

\mu_0(A) = \sum_{k = 1}^n \mu_0(A_k).

Then $\mu_0$ is a well-defined premeasure on $\mathcal A$ .

Note that, for an increasing function $F$ , we write

F(-\infty) = \lim_{x \to -\infty} F(x) \quad F(\infty) = \lim_{x \to \infty} F(x),

with the value possibly being $\pm \infty$ if the function diverges at the tails.

[1.5.3]Theorem(Lebesgue-Stieltjes Correspondence)#

Suppose $F: \bR \to \bR$ is increasing and right-continuous. Then there exists a unique Borel measure $\mu_F$ on $\bR$ such that $\mu_F((a, b]) = F(b) - F(a)$ for all $a, b \in \bR$ . If $G$ is another such function, we have that $\mu_F = \mu_G$ if and only if $F - G \equiv c$ for some constant $c \in \bR$ .

Conversely, if $\mu$ is some Borel measure on $\bR$ that is finite on all bounded Borel sets, we can define

F_k(x) = \begin{cases} \mu((k, x]) & x > k, \\ 0 & x = k, \\ -\mu((x, k]) & x < k \end{cases}

for any $k \in \bR$ . Then each $F_k$ is increasing, right-continuous, and such that $\mu = \mu_{F_k}$ .

Crucially, this correspondence only exists for Borel measures that are finite on all bounded sets; without this condition, the construction of a real-valued $F$ would immediately collapse. Even with this guarantee, one might naturally attempt to define the generating function globally as simply $F(x) = \mu((-\infty, x])$ . However, this strictly requires the measure to be finite on the unbounded ray $(-\infty, x]$ , which is not guaranteed for general boundedly-finite Borel measures (such as the standard Lebesgue measure, which is infinite on all such rays). By introducing a finite anchor point $k \in \bR$ , we restrict our evaluations entirely to bounded intervals. Because the measure is finite on bounded sets, this ensures $F_k(x)$ safely remains finite and well-defined everywhere.

Consequently, the correspondence very explicitly associates a measure with a family of functions that all differ by an additive constant. The theorem already dictates that any two functions generating the same Borel measure via the Lebesgue-Stieltjes construction must differ by a constant. Also, the constructed family of functions $F_k$ generating an arbitrary Borel measure differ by a constant as well. Indeed, without loss of generality, let us compare $F_k$ against the function $F_0$ anchored at $0$ . For any $x > k > 0$ , finite additivity yields

F_0(x) = \mu((0, x]) = \mu((0, k]) + \mu((k, x]) = F_0(k) + F_k(x).

Rearranging this gives $F_k(x) = F_0(x) - F_0(k)$ . It is quick to show this exact identity holds for all relative orderings of $x, k,$ and $0$ . Thus, we can readily recover one function from another; modifying the anchor point to $k$ simply translates the function $F_0$ vertically by exactly $F_0(k)$ .