**Infinite-dimensional Lebesgue Measure**

In mathematics, it is a theorem that **there is no analogue of Lebesgue measure on an infinite-dimensional Banach space**. Other kinds of measures are therefore used on infinite-dimensional spaces: often, the abstract Wiener space construction is used. Alternatively, one may consider Lebesgue measure on finite-dimensional subspaces of the larger space and consider so-called prevalent and shy sets.

Compact sets in Banach spaces may also carry natural measures: the Hilbert cube, for instance, carries the product Lebesgue measure. In a similar spirit, the compact topological group given by the Tychonoff product of infinitely many copies of the circle group is infinite-dimensional, and carries a Haar measure that is translation-invariant.

Read more about Infinite-dimensional Lebesgue Measure: Motivation, Statement of The Theorem, Proof of The Theorem

### Famous quotes containing the word measure:

“Perpetual modernness is the *measure* of merit, in every work of art; since the author of it was not misled by anything short- lived or local, but abode by real and abiding traits.”

—Ralph Waldo Emerson (1803–1882)