nLab Lebesgue space

Idea

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 $L^p$ space (or $L_p$ space). Here we discuss the latter.

Lebesgue spaces $L^p$ 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 ‘$L_p$’ is used as a synonym for $L^p$; sometimes it is used to mean $L^{1/p}$.

Notation

For historical reasons (starting with the original paper by Riesz), the exponent $p$ is traditionally taken to be the reciprocal of the “correct” exponent.

If we take $M^p=L^{1/p}$, the spaces $M^p$ form a $\mathbf{C}$-graded algebra, where $\mathbf{C}$ denotes complex numbers.

This is a conceptual explanation for the appearance of formulas like $1/p+1/q=1/r$ in Hölder's inequality.

In differential geometry, the notion of density does use the “correct” grading.

In the Tomita–Takesaki theory, the parameter $t$ for modular automorphism group is almost the “correct” grading, except that it is multiplied by the imaginary unit $i$.

Definitions

If $1 \leq p \lt \infty$ is a real number and $(\Omega,\mu)$ is a measure space, one considers the $L^p$ space $L_p(\Omega)$, which is the vector space of equivalence classes of those measurable (complex- or real-valued) functions $f\colon \Omega \to \mathbb{K}$ whose (absolute values of) $p$th powers are integrable, in that the integral

$${\|f\|_p} \;\coloneqq\; \left( \textstyle{\int}_\Omega {|f|^p} \,d\mu \right)^{1/p} \;\lt\; \infty$$

exists. Two such are taken to be equivalent, $f \sim g$, if ${\|f-g\|_p} = 0$. For $p = 2$ this is the space $L^2$ of square integrable functions.

On these spaces $L^p(X)$ of equivalence classes of $p$-power integrable functions, the function ${\|f\|_p}$ 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 $L^p$ spaces are examples of Banach spaces; they are continuous analogues of $l^p$ spaces of $p$-summable series. (Indeed, $l^p(S)$, for $S$ a set, is simply $L^p(S)$ if $S$ is equipped with counting measure.)

For fixed $f$, the norm ${\|f\|_p}$ is continuous in $p$. Accordingly, for $p = \infty$, one may take the limit of ${\|f\|}_p$ as $p \to \infty$. However, this turns out to be the same as the essential supremum norm $\|f\|_\infty$. Therefore, $L^\infty(\Omega)$ makes sense as long as $\Omega$ is a measurable space equipped with a family of null sets (or full sets); the measure $\mu$ is otherwise irrelevant.

For $0 \leq p \lt 1$, one can modify the definition to make $L^p$ into an F-space (but not a Banach space). See the definitions at p-norm.

Minkowski’s inequality

We offer here a proof that ${\|f\|_p}$ indeed defines a norm in the case $1 \lt p \lt \infty$, in that it satisfies the triangle inequality. This is usually known as Minkowski's inequality.

(The cases $p = 1$ and $p = \infty$ 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 $f$ may be assumed to be real- or complex-valued.

Theorem

Suppose $1 \leq p \leq \infty$, and suppose $\Omega$ is a measure space with measure $\mu$. Then the function ${|(-)|_p} \,\colon\, L^p(\Omega, \mu) \to \mathbb{R}$ defined by

$${\|f\|_p} \;\coloneqq\; \big( \textstyle{\int}_\Omega {|f|^p} \,d\mu \big)^{1/p}$$

defines a norm.

One must verify three things:

1. Separation axiom: ${\|f\|_p} = 0$ implies $f = 0$.

2. Scaling axiom: ${\|t f\|}_p = {|t|} \, {\|f\|_p}$.

3. Triangle inequality: ${\|f + g\|_p} \leq {\|f\|_p} + {\|g\|_p}$.

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 $x \mapsto {|x|^p}$ (as a function from real or complex numbers to nonnegative real numbers).

First, some generalities. Let $V$ be a (real or complex) vector space equipped with a function ${\|(-)\|}\colon V \to [0, \infty]$ that satisfies the scaling axiom: ${\|t v\|} = {|t|} \, {\|v\|}$ for all scalars $t$, and the separation axiom: ${\|v\|} = 0$ implies $v = 0$. As usual, we define the unit ball in $V$ to be $\{v \in V \;|\; {\|v\|} \leq 1\}.$

Lemma

Given that the scaling and separation axioms hold, the following conditions are equivalent:

1. The triangle inequality is satisfied.
2. The unit ball is convex.
3. If ${\|u\|} = {\|v\|} = 1$, then ${\|t u + (1-t)v\|} \leq 1$ for all $t \in [0, 1]$.

Proof

Condition 1. implies condition 2. easily: if $u$ and $v$ are in the unit ball and $0 \leq t \leq 1$, we have

$$\array{ {\|t u + (1-t)v\|} & \leq & {\|t u\|} + {\|(1-t)v\|} \\ & = & t {\|u\|} + (1-t) {\|v\|} \\ & \leq & t + (1-t) = 1.}$$

Now 2. implies 3. trivially, so it remains to prove that 3. implies 1. Suppose ${\|v\|}, {\|v'\|} \in (0, \infty)$. Let $u = \frac{v}{{\|v\|}}$ and $u' = \frac{v'}{{\|v'\|}}$ be the associated unit vectors. Then

$$\array{ \frac{v + v'}{{\|v\|}+{\|v'\|}} & = & (\frac{{\|v\|}}{{\|v\|}+{\|v'\|}})\frac{v}{{\|v\|}} + (\frac{{\|v'\|}}{{\|v\|}+{\|v'\|}})\frac{v'}{{\|v'\|}} \\ & = & t u + (1-t)u'}$$

where $t = \frac{{\|v\|}}{{\|v\|} + {\|v'\|}}$. If condition 3. holds, then

$${\|t u + (1-t)u'\|} \leq 1$$

but by the scaling axiom, this is the same as saying

$$\frac{{\|v + v'\|}}{{\|v\|} + {\|v'\|}} \leq 1,$$

which is the triangle inequality.

Consider now $L^p$ with its $p$-norm ${\|f\|} = {|f|_p}$. By Lemma , this inequality is equivalent to

• Condition 4: If ${|u|_{p}^{p}} = 1$ and ${|v|_{p}^{p}} = 1$, then ${|t u + (1-t)v|_{p}^{p}} \leq 1$ whenever $0 \leq t \leq 1$.

This allows us to remove the cumbersome exponent $1/p$ in the definition of the $p$-norm.

The next two lemmas may be proven by elementary calculus; we omit the proofs. (But you can also see the full details.)

Lemma

Let $\alpha, \beta$ be two complex numbers, and define

$$\gamma(t) = {|\alpha + \beta t|^p}$$

for real $t$. Then $\gamma''(t)$ is nonnegative.

Lemma

Define $\phi\colon \mathbb{C} \to \mathbb{R}$ by $\phi(x) = |x|^p$. Then $\phi$ is convex, i.e., for all $x, y$,

$${|t x + (1-t)y|^p} \leq t{|x|^p} + (1-t){|y|^p}$$

for all $t \in [0, 1]$.

Proof of Minkowski’s inequality

Let $u$ and $v$ be unit vectors in $L^p$. By condition 4, it suffices to show that ${|t u + (1-t)v|_p} \leq 1$ for all $t \in [0, 1]$. But

$$\textstyle{\int}_\Omega {|t u + (1-t)v|^p} \,d\mu \;\leq\; \textstyle{\int}_\Omega t{|u|}^p + (1-t){|v|}^p \,d\mu \,,$$

by Lemma . Using $\int {|u|^p} = 1 = \int {|v|^p}$, we are done.

Independence from the measure

In this section, we exhibit a construction of $L^p$-spaces that only makes use of an enhanced measurable space $(X,M,N)$ (a set $X$ with a σ-algebra $M$ of measurable sets and a σ-ideal $N$ of negligible sets), and not the full data of a measure space $(X,M,\mu)$ (where $\mu$ is a measure on $(X,M)$ that vanishes on $N$).

This makes it clear that $L^p(X,M,\mu)$ does not really depend on $\mu$, so we can also write $L^p(X,M,N)$ instead.

We set $\mathcal{L}^p = L^{1/p}$, to ensure we get a $\mathbf{C}$-graded algebra. In particular,

• $\mathcal{L}^0 = L^\infty$ (bounded measurable functions),

• $\mathcal{L}^{1/2}=L^2$ (the Hilbert space of half-densities),

• $\mathcal{L}^1=L^1$ (the space of finite measures).

Given $p\in\mathbf{C}$, we define $\mathcal{L}^p(X,M,N)$ as a quotient. The underlying set is $\mathcal{L}^0(X,M,N)\times \mathcal{L}^1(X,M,N)$, the set of (equivalence classes of) bounded measurable complex-valued functions on $(X,M,N)$ times the set of of finite complex-valued measures on $(X,M)$ that vanish on $N$.

The equivalence relation is defined as follows: $(f,g\cdot\mu)\sim(f g^p,\mu)$, for every nonnegative measurable function $g$ on $(X,M,N)$. Here $g\cdot\mu$ is a unique measure supplied by applying the Radon–Nikodym theorem in reverse:

$$(g\cdot\mu)(A) = \int_A g d\mu.$$

For $p=1$, the map $(f,\mu)\mapsto f\cdot \mu$ establishes an isomorphism from $\mathcal{L}^1(X,M,N)$ to the vector space of finite complex-valued measures on $(X,M,N)$.

For $p=0$, the map $(f,\mu)\mapsto f supp(\mu)$ establishes an isomorphism from $\mathcal{L}^0(X,M,N)$ to the vector space of bounded complex-valued functions on $(X,M,N)$.

Given a measure space $(X,M,\mu)$, the map $f \mapsto f \mu^p$ establishes an isomorphism

$$\mathcal{L}^p(X,M,\mu)\to\mathcal{L}^p(X,M,N),$$

where the left side is the usual $L^p$-space from analysis (with the reciprocal index $p$).

Related concepts

• square integrable function

• canonical Hilbert space of half-densities

• tempered distribution

References

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

category: analysis