14  Lebesgue spaces

As claimed, the theory of PDEs are formulated using function spaces. These function spaces are in general called Lebesgue spaces, and are based on the Lebesgue integral.

In this section, let \((X,\boldsymbol{X},\mu)\) be a fixed measure space, for example a measurable subset \(\Omega\subset \mathbb{R}^n\) with Lebesgue measure, and we consider \(\mathbb{F}\in \{\mathbb{R}, \mathbb{C}\}\) a measure space with the Lebesgue measure. For complex functions, we consider really \(\mathbb{C}\) as \(\mathbb{R}^2\).

The reader may have encountered square integrable functions before, i.e., functions \(f : X \to \mathbb{C}\) that satisfies \[\int_\Omega |f(x)|^2 \, \mathrm{d}x < +\infty.\] Here, \(\mathrm{d}x\) is short for the Lebesgue measure. The reader may also be used to thinking of \(f\) as a point in Hilbert space \(L^2(\Omega)\). However, it is not hard to see, that we can have \(f \neq g\) but \[\int |f-g|^2 \; \mathrm{d}^n x = 0 \quad !\] So, are the functions really different?

In quantum mechanics, the wavefunction \(\psi(x)\) fo a particle defines a probability density \(P(x) = |\psi(x)|^2\). The theory of probability also work with measurable spaces, and the probability of locating the particle in a subset \(A \subset \Omega\) is given by \(\int_A P(x) \,\mathrm{d}x\), where \(A\) is measurable. The pointwise definition of a probability density is therefore only meaningful up to a set of measure zero, i.e., “almost everywhere”.

Let \(f : X \to \mathbb{F}\) be a measurable function. Recall that \(\int_X f\, \mathrm{d}\mu=0\) if and only if \(f(x)=0\) “almost everywhere”. An example is the function over \(\mathbb{R}^n\) which is zero except for at the points with rational coordinates. As the integral concerned, this function is zero!

To define function spaces with integrals involved in norms and inner products rigorously, we need the following:

Definition 14.1 (Equivalence classes of measurable functions) Let \(f,g : X \to \mathbb{F}\) be measurable functions. We say that \(f\) and \(g\) are equivalent, written \(f\sim g\), if \(f=g\) almost everywhere. We write \[[f] = \{ g \mid f\sim g\}\] for the equivalence class of functions that are almost everywhere the same.

Definition 14.2 (Lebesgue spaces) Let \(p\geq 1\), and define the \(p\)-norm \[\|[f]\|_p = \|f\|_p = \left( \int_X |f(x)|^p \, \mathrm{d}^n x\right)^{1/p}\] The space \(L^p(X)\) is defined as \[L^p(X) = \left\{ [f] \mid f : X \to \mathbb{F}\;\text{measurable},\; \|f\|_p < +\infty \right\}.\]

Theorem 14.1 (Statement) The Lebesgue spaces \(L^p(X)\) are complete normed spaces, i.e., Banach spaces. (See Functional analysis.)

The space \(L^2(X)\) is a Hilbert space with inner product \[\left\langle f,g\right\rangle_{L^2(X)} = \int \overline{f(x)} g(x) \; \mathrm{d}^n x.\]

The fact that \(L^p\) spaces are complete is of great importance. It guarantees that if we consider Cauchy sequences in these spaces, they are converge to something in the space. This is extremely useful when studying PDEs.

To close this section, we demonstrate just how general the Lebesgue spaces are, and how “wild” integrable functions can be.

Example 14.1 (A wild function) Let \(X = \mathbb{R}^3\) with Lebesgue measure. Let \(u(\mathbf{x}) = \|\mathbf{x}\|^{-1} e^{-\|\mathbf{x}\|}\). This function is square integrable. (Can you prove it?) The function is also unbounded as \(\|\mathbf{x}\|\to 0\).

Now let \(\mathbb{Q}^3\) be the set of rational coordinates in \(\mathbb{R}^3\). It is a countable set, so we may write it as a sequence \(\mathbf{y}_i\), \(i \in \mathbb{N}\). Consider the function \[f(\mathbf{x}) = \sum_{i=1}^\infty 2^{-i} u(\mathbf{x}-\mathbf{y}_i),\] i.e., at every rational point, we place a singularity.

It now follows, that for every \(\epsilon\)-ball in \(\mathbb{R}^3\), no matter how small, the function is unbounded. Yet, \[\|f\|_{L^2} \leq \sum_{i=1}^\infty 2^{-i} \|u\|_{L^2} < +\infty.\] Not only is the norm finite, but since \(L^2\) is complete, the series actually converges to an element in \(L^2(X)\). This function is unbounded in every arbitrarily small region.