Measure Theory 13 | Lebesgue-Stieltjes Measures [dark version]

Q: Why can’t μ_F((a, b]) always be computed as F(b)−F(a) for a monotone F?

Because monotone functions can jump. In (a, b], the point b is excluded, so the relevant contribution at b uses the left limit F(b−). The point a is included, so the relevant contribution at a uses the right limit F(a+). That yields μ_F((a, b]) = F(b−) − F(a+), not F(b)−F(a) when discontinuities occur.

Q: What does it mean that the measure “doesn’t care” about the value of F at a jump point?

If F has a discontinuity at x0, changing F(x0) without changing the one-sided limits F(x0−) and F(x0+) leaves μ_F unchanged. The interval measures depend only on the jump size captured by those one-sided limits, so the exact function value at the point itself never enters the construction.

Q: How does the construction recover ordinary Lebesgue measure?

Set F(x)=x. Then F is continuous and increasing with no jumps, so F(b−)=F(b) and F(a+)=F(a). The interval measure becomes μ_F((a, b]) = b−a, which matches standard interval length and therefore gives Lebesgue measure on Borel sets.

Q: How does a step function produce a Dirac measure?

Let F(x)=0 for x 0. By uniqueness of the extension, this matches the Dirac (point-mass) measure at 0.

TL;DR

Lebesgue–Stieltjes measures convert any non-decreasing function F on ℝ into a measure μ_F by weighting interval lengths according to increases in F.

Briefing Cornell Notes

Briefing

Lebesgue–Stieltjes measures turn any non-decreasing function on the real line into an ordinary measure, letting “interval length” be weighted by how that function accumulates. The construction starts with a monotone function F and defines the measure of half-open intervals (a, b] by looking at how much F increases across the interval—carefully accounting for jumps using left and right limits. This matters because it generalizes standard length: when F(x)=x, the resulting measure is exactly Lebesgue measure, while other choices of F produce measures concentrated on points or spread according to a prescribed “accumulation profile.”

The key technical step is handling discontinuities. For a monotone F, jumps can occur, so the measure of (a, b] is not simply F(b)−F(a) when F has a jump at a or b. Since b is excluded, the right endpoint contributes using the left limit at b, written F(b−). Since a is included, the left endpoint contributes using the right limit at a, written F(a+). With this convention, the measure of (a, b] becomes

μ_F((a, b]) = F(b−) − F(a+).

Equivalently, the construction can be phrased using the appropriate one-sided limits: the size of any jump depends only on the left and right limits, not on the value of F at the jump point itself. That observation is crucial: changing F at a single point does not change the resulting measure.

Once μ_F is defined on these half-open intervals, a measure-extension theorem (the Carathéodory extension theorem, referenced via the “semi-ring” property of such intervals) extends the definition uniquely to the full Borel σ-algebra on ℝ. In other words, specifying μ_F on intervals of the form (a, b] pins down exactly one measure on all Borel sets, denoted μ_F. This uniqueness is what makes the approach powerful: define it on a simple generating class, and the rest follows.

Several examples clarify the mechanism. Taking F(x)=x yields μ_F((a, b])=b−a, recovering ordinary Lebesgue measure. Choosing a constant function F(x)=1 produces μ_F((a, b])=0 for all intervals, giving the zero measure. A step function illustrates how jumps create point-mass behavior: if F(x)=0 for x<0 and F(x)=1 for x≥0, then any interval containing 0 in the right way—like (−ε, ε]—has measure 1, matching the Dirac measure at 0. The construction ignores the exact value of F at x=0, since only the one-sided limits determine the jump.

Finally, when F is continuously differentiable and non-decreasing (so F′ is continuous and non-negative), there are no jumps. The measure of (a, b] simplifies to F(b)−F(a), and by the fundamental theorem of calculus this equals ∫_a^b F′(x) dx. In that setting, μ_F can be defined for Borel sets via integration against the “density” F′, showing how Lebesgue–Stieltjes measures connect to weighted Lebesgue measures and density functions.

Cornell Notes

Lebesgue–Stieltjes measures build a measure μ_F from any non-decreasing function F on ℝ. For half-open intervals (a, b], the measure is determined by how F accumulates across the interval using one-sided limits: μ_F((a, b]) = F(b−) − F(a+). Jumps are handled automatically, and the measure does not depend on the value of F at a discontinuity—only the left and right limits matter. Defining μ_F on these intervals is enough: Carathéodory’s extension theorem gives a unique measure on the Borel σ-algebra. When F is C¹ with F′≥0, μ_F((a, b]) becomes ∫_a^b F′(x) dx, so F′ acts like a density.

Why can’t μ_F((a, b]) always be computed as F(b)−F(a) for a monotone F?

Because monotone functions can jump. In (a, b], the point b is excluded, so the relevant contribution at b uses the left limit F(b−). The point a is included, so the relevant contribution at a uses the right limit F(a+). That yields μ_F((a, b]) = F(b−) − F(a+), not F(b)−F(a) when discontinuities occur.

What does it mean that the measure “doesn’t care” about the value of F at a jump point?

If F has a discontinuity at x0, changing F(x0) without changing the one-sided limits F(x0−) and F(x0+) leaves μ_F unchanged. The interval measures depend only on the jump size captured by those one-sided limits, so the exact function value at the point itself never enters the construction.

How does the construction recover ordinary Lebesgue measure?

Set F(x)=x. Then F is continuous and increasing with no jumps, so F(b−)=F(b) and F(a+)=F(a). The interval measure becomes μ_F((a, b]) = b−a, which matches standard interval length and therefore gives Lebesgue measure on Borel sets.

How does a step function produce a Dirac measure?

Let F(x)=0 for x<0 and F(x)=1 for x≥0. For intervals entirely left of 0 or entirely right of 0, the one-sided limits give F(b−)=F(a+) so μ_F((a, b])=0. For intervals that straddle 0 in the right way—e.g., (−ε, ε]—the measure becomes 1−0=1 for any ε>0. By uniqueness of the extension, this matches the Dirac (point-mass) measure at 0.

When F is smooth, how does μ_F relate to an integral of a density?

If F is continuously differentiable and non-decreasing, then F has no jumps and μ_F((a, b]) = F(b)−F(a). By the fundamental theorem of calculus, F(b)−F(a)=∫_a^b F′(x) dx. This identifies F′ as the density driving the Lebesgue–Stieltjes measure.

Review Questions

Given a monotone function F with a jump at x0, which one-sided limits determine μ_F((a,b]) when a=x0 or b=x0?
Compute μ_F((a,b]) for F(x)=x and for F(x)=1, using the interval formula with one-sided limits.
For F(x)=0 for x<0 and F(x)=1 for x≥0, what is μ_F((−ε,ε]) and why does the exact value at x=0 not matter?

Key Points

1
Lebesgue–Stieltjes measures convert any non-decreasing function F on ℝ into a measure μ_F by weighting interval lengths according to increases in F.
2
For half-open intervals (a, b], the correct formula is μ_F((a, b]) = F(b−) − F(a+), which uses left and right limits to handle jumps.
3
The measure depends only on the one-sided limits of F at discontinuities; changing F at a single point does not change μ_F.
4
Defining μ_F on the semi-ring of half-open intervals (a, b] uniquely determines μ_F on all Borel sets via Carathéodory’s extension theorem.
5
When F(x)=x, μ_F equals Lebesgue measure because μ_F((a,b])=b−a.
6
A step function for F produces point-mass behavior; for example, F(x)=0 (x<0) and 1 (x≥0) yields the Dirac measure at 0.
7
If F is C¹ and non-decreasing, then μ_F((a,b])=∫_a^b F′(x) dx, so F′ acts as a density.

Highlights

The interval formula μ_F((a, b]) = F(b−) − F(a+) is the entire mechanism for handling jumps in monotone functions.

Lebesgue–Stieltjes measures ignore the value of F at discontinuity points; only F(x−) and F(x+) matter.

With F(x)=x, the construction collapses to ordinary Lebesgue measure.

A single jump in F can generate a Dirac measure, concentrating all mass at one point.

For smooth F, the measure becomes an integral against the derivative F′, linking the construction to density functions.

Topics

Lebesgue–Stieltjes Measures
Monotone Functions
One-Sided Limits
Carathéodory Extension
Density via Derivative