---
tag: 23DM2PQC78
kind: statement
role: theorem
name: Nonexistence of infinite-dimensional Lebesgue measure
msc: [28C20, 46G12]
uses: [06GZ956657, DYR1QFNGNJ, 2Z2QVWZ262, 097C7KCAT5, 0DC3MPR7C9]
about: [06GZ956657, DYR1QFNGNJ]
sources:
  - url: https://en.wikipedia.org/wiki/Infinite-dimensional_Lebesgue_measure
---

## Statement

Let $X$ be an infinite-dimensional separable real [[Banach space|06GZ956657]].
Let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$ such that these conditions hold:

1. $\mu$ is translation-invariant: $\mu(x + A) = \mu(A)$ for every Borel set $A$ and every $x \in X$.
2. $\mu$ is locally finite: every point of $X$ has an open neighbourhood of finite measure.

Then $\mu(A) = 0$ for every Borel set $A$.

## Remarks

By clause 1 and clause 2, some [[open ball|0DC3MPR7C9]] $B(0, r)$ with $r > 0$ has finite measure.
By [[Riesz's lemma|097C7KCAT5]], there are infinitely many points $x_1, x_2, \ldots$ with $\norm{x_i} = 3r/4$ and $\norm{x_i - x_j} \ge r/2$ for $i \ne j$.
The balls $B(x_i, r/4)$ are disjoint, lie in $B(0, r)$, and all have the same measure by translation invariance; so this measure is $0$.
By separability, $X$ is a countable union of balls of radius $r/4$, and so $\mu(X) = 0$.

The result is often stated in an equivalent way: a translation-invariant Borel measure that is not identically zero gives infinite measure to every nonempty open set.
In finite dimension the conclusion fails, because [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$ is translation-invariant and locally finite.
Gaussian measures on infinite-dimensional spaces avoid the theorem by giving up translation invariance.

## Explanations

### L1 human

In $\R^n$ volume does not change when a set is moved, and small balls have finite volume.
In infinite dimension no notion of volume can have both properties, except the one that gives volume zero to everything.
The reason is that a ball contains infinitely many disjoint smaller balls of a fixed size, which would all need the same positive volume.

### L2 human

The theorem explains why integration over spaces of functions, as in probability theory and quantum field theory, uses other measures.
The standard choices are Gaussian measures, such as Wiener measure on paths.
Their behaviour under translations is described by the Cameron–Martin theorem: only translations along a small subspace keep the measure class.