Measure and integration Signed measures and differentiation of measures Theorem

Vitali–Hahn–Saks theorem

Builds on Measure Finite measure Complex measure Absolutely continuous measure

Statement

Let $(X, \Sigma, \lambda)$ be a measure space with $\lambda$ a finite measure. Let $(\mu_n)_{n \ge 1}$ be a sequence of complex measures on $\Sigma$ , each absolutely continuous with respect to $\lambda$ . Assume that for every $E \in \Sigma$ the limit $\mu(E) = \lim_{n \to \infty} \mu_n(E)$ exists in $\C$ . Then:

The $\mu_n$ are uniformly absolutely continuous with respect to $\lambda$ : for every $\varepsilon > 0$ there is $\delta > 0$ such that $\lambda(E) < \delta$ implies $\abs{\mu_n(E)} < \varepsilon$ for all $n$ .
The set function $\mu$ is a complex measure on $\Sigma$ , and it is absolutely continuous with respect to $\lambda$ .

Remarks

A standard proof uses the Baire category theorem. The measurable sets modulo $\lambda$ -null sets form a complete metric space with distance $d(E, F) = \lambda(E \,\triangle\, F)$ . Each $\mu_n$ is continuous on this space, and a Baire argument on the sets where $\abs{\mu_n(E) - \mu_m(E)} \le \varepsilon$ for all $m, n \ge N$ gives the uniform bound.

Example: let $f_n \in L^1(\lambda)$ with $\int_E f_n \, d\lambda$ convergent for every $E$ . The theorem applied to $\mu_n = f_n \lambda$ shows that $(f_n)$ is uniformly integrable, which is a key step in the Dunford–Pettis theorem.

The hypothesis asks for convergence on every set of $\Sigma$ . Convergence on a generating family of sets, such as intervals, is not enough in general.

Going deeper

For a student

The theorem is the measure version of the uniform boundedness principle: pointwise convergence on a complete space forces a uniform estimate. It is used to characterise weakly convergent and weakly compact sets in $L^1$ . The Nikodym convergence theorem is a version that needs no reference measure $\lambda$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`N0K0YWPQN6`
URI	`m2://N0K0YWPQN6`
Kind	statement, theorem
Writing	W0 stub
MSC	28A33, 28A10
Status	This copy of the site has no build data. Run `m2 status N0K0YWPQN6`.
About	Absolutely continuous measure
Uses	Measure, Finite measure, Complex measure, Absolutely continuous measure
Same as	Wikidata Q7936914 Wikipedia Vitali–Hahn–Saks theorem
Changed	2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

an inventory row: the row of the inventory that the node accounts for (0 of 1)
a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.