Motivation
It can be shown that Lebesgue measure λn on Euclidean space Rn is locally finite, strictly positive and translation-invariant, explicitly:
- every point x in Rn has an open neighbourhood Nx with finite measure λn(Nx) < +∞;
- every non-empty open subset U of Rn has positive measure λn(U) > 0; and
- if A is any Lebesgue-measurable subset of Rn, Th : Rn → Rn, Th(x) = x + h, denotes the translation map, and (Th)∗(λn) denotes the push forward, then (Th)∗(λn)(A) = λn(A).
Geometrically speaking, these three properties make Lebesgue measure very nice to work with. When we consider an infinite-dimensional space such as an Lp space or the space of continuous paths in Euclidean space, it would be nice to have a similarly nice measure to work with. Unfortunately, this is not possible.
Read more about this topic: Infinite-dimensional Lebesgue Measure
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