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The Stieltjes interval set function is finitely additive on the half-open interval algebra

Statement

Let $F : R \to R$ be nondecreasing and right-continuous, and let $μ_{0, F}$ be the set function of The interval set function attached to a nondecreasing right-continuous function. Then the value of $μ_{0, F} (E)$ is independent of the chosen finite disjoint decomposition of $E \in H$ . Moreover, if $E, G \in H$ are disjoint, then

$μ_{0, F} (E \cup G) = μ_{0, F} (E) + μ_{0, F} (G) .$

So $μ_{0, F}$ is a finitely additive set function on the half-open interval algebra.

Facts & Assumptions

Given: A nondecreasing right-continuous function $F : R \to R$ and the set function $μ_{0, F}$ defined from finite disjoint unions of h-intervals.

[L1]

For every disjoint presentation $E = ⋃_{i = 1}^{m} I_{i}$ in the half-open interval algebra, the proposed value is $μ_{0, F} (E) = \sum_{i = 1}^{m} μ_{0, F} (I_{i})$ . (The interval set function attached to a nondecreasing right-continuous function)

Proof

technique · direct

1.1

Suppose $E = ⋃_{i = 1}^{m} I_{i} = ⋃_{j = 1}^{n} J_{j}$ are two finite disjoint h-interval decompositions of the same set.

givenL1

Collect every finite endpoint appearing among the $I_{i}$ and $J_{j}$ , and adjoin $- \infty$ or $+ \infty$ when a left or right ray occurs. This gives an increasing list $p_{0} < \dots < p_{r}$ such that each $I_{i}$ and each $J_{j}$ is a disjoint union of consecutive h-cells $C_{ℓ} : = (p_{ℓ - 1}, p_{ℓ}]$ , with the first or last cell possibly a ray.

2.1

If $I$ is one interval from either decomposition, the sum of the $μ_{0, F}$ -values of the consecutive cells inside $I$ telescopes to $μ_{0, F} (I)$ .

step 1.1L1algebra

In the unbounded cases this is exactly the truncation/supremum convention built into The interval set function attached to a nondecreasing right-continuous function. Thus each decomposition gives the same total, namely the sum of the cell values over those $C_{ℓ}$ contained in $E$ . So $μ_{0, F} (E)$ is well defined. [step 1.1, L1, algebra]

3.1

Let $E, G \in H$ be disjoint, and choose disjoint h-interval decompositions of $E$ and of $G$ .

step 2.1givenalgebra

Refine them to a common endpoint grid as in step 1.1, and sum over the grid cells. The cells belonging to $E \cup G$ are exactly the disjoint union of the cells belonging to $E$ and the cells belonging to $G$ , so the corresponding cell sums add:

$μ_{0, F} (E \cup G) = μ_{0, F} (E) + μ_{0, F} (G) .$

Together with step 2.1 this proves the claim. [step 2.1, given, algebra] ∎

Depends on

The interval set function attached to a nondecreasing right-continuous function

Used by

The Stieltjes interval set function is a premeasure Theorem

Cited to discharge well-definedness by The interval set function attached to a nondecreasing right-continuous function.

Dependency tree · two levels

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Sources

Gerald B. Folland, Real Analysis, 2nd ed., Proposition 1.15 (standard reference, not scraped)