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The distribution function of a Borel measure on $R$ , normalized at $0$

Definition

Let $μ$ be a Borel measure on $R$ finite on compact sets. Its distribution function normalized at $0$ is the function $F_{μ} : R \to R$ defined by

$F_{μ} (x) : = {\begin{cases} μ ((0, x]), & x \geq 0, \\ - μ ((x, 0]), & x < 0. \end{cases}$

Both interval measures are finite: the relevant half-open interval is contained in the closed bounded interval with endpoints $0$ and $x$ , which is compact by Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact, and measure monotonicity Measures are monotone applies.

The two cases agree at $x = 0$ , where both give $0$ . This normalization removes the additive-constant ambiguity that would remain if one used only interval increments.

Depends on

Used by

Assuming countable choice, finite-on-compacts Borel measures on ℝ correspond to nondecreasing right-continuous functions modulo constants Theorem

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Sources

Gerald B. Folland, Real Analysis, 2nd ed., Section 1.5 (standard reference, not scraped)
John K. Hunter, Measure Theory, Section 2.9 (standard reference, not scraped)

The distribution function of a Borel measure on RR, normalized at 00