Infinite-dimensional Lebesgue measure
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An infinite-dimensional Lebesgue measure is a type of measure defined on an infinite-dimensional Banach space. It shares properties with the Lebesgue measure on finite-dimensional spaces.
However, the usual Lebesgue measure cannot be extended to all infinite-dimensional spaces. This limitation arises because any translation-invariant Borel measure on an infinite-dimensional separable Banach space is always either infinite for all sets or zero for all sets. Despite this, there are some instances of Lebesgue-like measures. These occur when the space is not separable, such as the Hilbert cube, or when one of the key properties of the Lebesgue measure is relaxed.
Motivation
The Lebesgue measure on the Euclidean space is locally finite, strictly positive, and translation-invariant. That is:
- every point in has an open neighborhood with finite measure:
- every non-empty open subset of has positive measure: and
- if is any Lebesgue-measurable subset of and is a vector in then all translates of have the same measure:
Motivated by their geometrical significance, constructing measures satisfying the above set properties for infinite-dimensional spaces such as the spaces or path spaces[disambiguation needed] is still an open and active area of research.
Statement of the theorem
On a non locally compact
This theorem implies that on an infinite dimensional separable Banach space - which can not be locally compact - a Lebesgue measure does not exist.
Non-Existence Theorem in Separable Banach spaces
Let be an infinite-dimensional, separable Banach space. Then, the only locally finite and translation invariant Borel measure on is the trivial measure. Equivalently, there is no locally finite, strictly positive, and translation invariant measure on .[2]
Proof
Let be an infinite-dimensional, separable Banach space equipped with a locally finite translation-invariant measurement To prove that is the trivial measure, it is sufficient and necessary to show that
Like every separable metric space, is a Lindelöf space, which means that every open cover of has a countable subcover. It is, therefore, enough to show that there exists some open cover of by null sets because by choosing a countable subcover, the σ-subadditivity of implies that
Using local finiteness, suppose that for some the
Since was arbitrary, every open ball in has zero measure, and taking a cover of which is the set of all open balls completes the proof.
Nontrivial measures
Here are some examples of infinite-dimensional Lebesgue measures that can exist if the conditions of the above theorem are relaxed.
There are other
The
See also
- Cylinder set measure – way to generate a measure over product spaces
- Cameron–Martin theorem – Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.
- Feldman–Hájek theorem – Theory in probability theory
- Gaussian measure#Infinite-dimensional spaces – Type of Borel measure
- Structure theorem for Gaussian measures – Mathematical theorem
- Projection-valued measure – Mathematical operator-value measure of interest in quantum mechanics and functional analysis
- Set function – Function from sets to numbers
References
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- S2CID 17534021.)
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