Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
The term ‘Lebesgue space’ can stand for two distinct notions: one is the general notion of measure space (compare the Springer Encyclopaedia of Mathematics) and another is the notion of space (or space). Here we discuss the latter.
Lebesgue spaces in this sense are normed vector spaces of functions on a measure space, equipped with the suitable version of the p-norm.
Beware that sometimes the notation ‘’ is used as a synonym for ; sometimes it is used to mean .
For historical reasons (starting with the original paper by Riesz), the exponent is traditionally taken to be the reciprocal of the “correct” exponent.
If we take , the spaces form a -graded algebra, where denotes complex numbers.
This is a conceptual explanation for the appearance of formulas like in Hölder's inequality.
In differential geometry, the notion of density does use the “correct” grading.
In the Tomita–Takesaki theory, the parameter for modular automorphism group is almost the “correct” grading, except that it is multiplied by the imaginary unit .
If is a real number and is a measure space, one considers the space , which is the vector space of equivalence classes of those measurable (complex- or real-valued) functions whose (absolute values of) th powers are integrable, in that the integral
exists. Two such are taken to be equivalent, , if . For this is the space of square integrable functions.
On these spaces of equivalence classes of -power integrable functions, the function satisfies the triangle inequality (due to Minkowski's inequality, see below) and hence defines a norm, the p-norm, making them normed vector spaces.
The spaces are examples of Banach spaces; they are continuous analogues of spaces of -summable series. (Indeed, , for a set, is simply if is equipped with counting measure.)
For fixed , the norm is continuous in . Accordingly, for , one may take the limit of as . However, this turns out to be the same as the essential supremum norm . Therefore, makes sense as long as is a measurable space equipped with a family of null sets (or full sets); the measure is otherwise irrelevant.
For , one can modify the definition to make into an F-space (but not a Banach space). See the definitions at p-norm.
We offer here a proof that indeed defines a norm in the case , in that it satisfies the triangle inequality. This is usually known as Minkowski's inequality.
(The cases and follow by continuity and are easy to check from first principles.)
The most usual textbook proofs involve a clever application of Hölder's inequality; the following proof is more straightforwardly geometric. All functions may be assumed to be real- or complex-valued.
Suppose , and suppose is a measure space with measure . Then the function defined by
defines a norm.
One must verify three things:
Separation axiom: implies .
Scaling axiom: .
Triangle inequality: .
The first two properties are obvious, so it remains to prove the last, which is also called Minkowski's inequality.
Our proof of Minkowski’s inequality is broken down into a series of simple lemmas. The plan is to boil it down to two things: the scaling axiom, and convexity of the function (as a function from real or complex numbers to nonnegative real numbers).
First, some generalities. Let be a (real or complex) vector space equipped with a function that satisfies the scaling axiom: for all scalars , and the separation axiom: implies . As usual, we define the unit ball in to be
Given that the scaling and separation axioms hold, the following conditions are equivalent:
Condition 1. implies condition 2. easily: if and are in the unit ball and , we have
Now 2. implies 3. trivially, so it remains to prove that 3. implies 1. Suppose . Let and be the associated unit vectors. Then
where . If condition 3. holds, then
but by the scaling axiom, this is the same as saying
which is the triangle inequality.
Consider now with its -norm . By Lemma , this inequality is equivalent to
This allows us to remove the cumbersome exponent in the definition of the -norm.
The next two lemmas may be proven by elementary calculus; we omit the proofs. (But you can also see the full details.)
Let be two complex numbers, and define
for real . Then is nonnegative.
Define by . Then is convex, i.e., for all ,
for all .
Let and be unit vectors in . By condition 4, it suffices to show that for all . But
Named after Henri Lebesgue.
W. Rudin, Functional analysis, McGraw Hill 1991.
L. C. Evans, Partial differential equations, Amer. Math. Soc. 1998.
Wikipedia (English): Lp space
Last revised on July 9, 2023 at 07:59:04. See the history of this page for a list of all contributions to it.