Separability on Sigma-Finite Countably Generated Spaces

Why Separability Requires Two Kinds of Control

A metric space is separable if it has a countable dense subset. For \(L^p\), density means that every equivalence class can be approximated in the \(L^p\) norm by members of that subset. Countably many generators provide a countable supply of measurable sets, but their indicators need not belong to \(L^p\) when the measure space has infinite measure. Sigma-finiteness addresses this difficulty: it lets us restrict attention to an increasing sequence of sets of finite measure, where indicators do have finite \(L^p\) norm.

Throughout, let \((X,\mathcal{M},\mu)\) be a measure space, not assumed complete, and fix \(1\le p<\infty\). Functions are real-valued and measurable, and \(L^p(\mu)\) consists of equivalence classes under equality \(\mu\)-a.e. of functions \(f\) satisfying \(\int_X |f|^p\,d\mu<\infty\). Sigma-finiteness means that \(X\) is a countable union of measurable sets of finite measure. We say \(\mathcal{M}\) is countably generated modulo null sets if there is a countable family \(\mathcal{G}\subseteq\mathcal{M}\) such that for every \(A\in\mathcal{M}\), some \(B\in\sigma(\mathcal{G})\) satisfies \(\mu(A\triangle B)=0\).

Approximating Sets Inside a Finite-Measure Region

Let \(\mathcal{A}\) be the algebra generated by \(\mathcal{G}\): it contains \(X\) and is closed under complements and finite unions. Since its elements are finite Boolean combinations of members of a countable family, \(\mathcal{A}\) is countable. The key point is that, on any fixed finite-measure region, its members approximate all measurable sets in symmetric-difference measure.

Proof. First take \(A\in\sigma(\mathcal{G})\). Let \(\mathcal{C}\) be the class of sets \(A\in\sigma(\mathcal{G})\) for which the stated approximation holds for every \(\varepsilon>0\). Each \(G\in\mathcal{G}\) belongs to \(\mathcal{C}\), because \(G\in\mathcal{A}\) gives zero error. If \(A\in\mathcal{C}\), choose \(B\in\mathcal{A}\) approximating \(A\cap F\). Then \(B^c\in\mathcal{A}\), and \[ (A^c\cap F)\triangle(B^c\cap F)=(A\cap F)\triangle(B\cap F). \] Thus \(\mathcal{C}\) is closed under complements.

For finite unions, if \(A_1,\dots,A_k\in\mathcal{C}\), choose \(B_i\in\mathcal{A}\) with \(\mu((A_i\cap F)\triangle(B_i\cap F))<\varepsilon/k\). The symmetric difference between the two unions is contained in the union of these \(k\) symmetric differences, so \(\bigcup_{i=1}^kB_i\) approximates \(\bigcup_{i=1}^kA_i\) with error less than \(\varepsilon\). Now suppose \(A_i\in\mathcal{C}\) for every \(i\ge1\), and put \(A=\bigcup_{i\ge1}A_i\). The sets \(F\cap\bigcup_{i=1}^k A_i\) increase to \(F\cap A\). Continuity from below of measure therefore gives a \(k\) for which \[ \mu\left((F\cap A)\setminus\left(F\cap\bigcup_{i=1}^kA_i\right)\right)<\varepsilon/2. \] Approximate the finite union \(\bigcup_{i=1}^k A_i\) by a member of \(\mathcal{A}\) with error less than \(\varepsilon/2\). The triangle inequality for symmetric-difference measure gives total error less than \(\varepsilon\). Hence \(\mathcal{C}\) is a sigma-algebra containing \(\mathcal{G}\), so it contains \(\sigma(\mathcal{G})\).

Finally, for a general \(A\in\mathcal{M}\), choose \(A'\in\sigma(\mathcal{G})\) with \(\mu(A\triangle A')=0\). Then \(\mu((A\cap F)\triangle(A'\cap F))=0\). The approximation just proved for \(A'\) also approximates \(A\), proving the lemma. \(\square\)

Proof of Separability

By sigma-finiteness, choose an increasing sequence \(F_1\subseteq F_2\subseteq\cdots\) of measurable sets with \(\mu(F_n)<\infty\) and \(\bigcup_nF_n=X\). For example, take finite-measure sets covering \(X\) and replace each by the union of the first \(n\). Define \[ \mathcal{D}=\left\{\sum_{j=1}^k q_j\chi_{F_n\cap B_j}: n,k\in\mathbb{N},\ q_j\in\mathbb{Q},\ B_j\in\mathcal{A}\right\}. \] Every member belongs to \(L^p(\mu)\), because it is bounded and supported on \(F_n\), which has finite measure. The set \(\mathcal{D}\) is countable: there are countably many choices of \(n\), and only countably many finite lists of rational coefficients and members of \(\mathcal{A}\).

To prove density, take \(f\in L^p(\mu)\) and \(\varepsilon>0\). Since \(F_n\uparrow X\), the Monotone Convergence Theorem applied to \(|f|^p\chi_{F_n}\) gives \[ \int_{F_n}|f|^p\,d\mu \longrightarrow \int_X|f|^p\,d\mu. \] Consequently, for sufficiently large \(n\), \(\|f-f\chi_{F_n}\|_p<\varepsilon/3\). The density of finite-measure-support simple functions in finite-\(p\) \(L^p\), established in the earlier material on density of simple functions, gives a simple function \(\varphi\) with \(\|f\chi_{F_n}-\varphi\|_p<\varepsilon/3\). Replacing \(\varphi\) by \(\varphi\chi_{F_n}\) cannot increase this error, since \(f\chi_{F_n}=0\) off \(F_n\). Write the resulting function as \(\sum_{j=1}^k c_j\chi_{A_j}\), where the \(A_j\) are disjoint measurable subsets of \(F_n\).

For each coefficient choose \(q_j\in\mathbb{Q}\) sufficiently close to \(c_j\). Because \(\mu(A_j)\le\mu(F_n)<\infty\), the \(L^p\) error from replacing \(c_j\) by \(q_j\) can be made arbitrarily small: \(\|(c_j-q_j)\chi_{A_j}\|_p=|c_j-q_j|\mu(A_j)^{1/p}\). For each \(j\), the lemma supplies \(B_j\in\mathcal{A}\) such that \(\mu(A_j\triangle(F_n\cap B_j))\) is as small as desired. The corresponding indicator error is \[ \|\chi_{A_j}-\chi_{F_n\cap B_j}\|_p =\mu(A_j\triangle(F_n\cap B_j))^{1/p}. \] There are only finitely many terms, so choose the rational coefficients and these set approximations to make the sum of all coefficient and set errors less than \(\varepsilon/3\). The triangle inequality then gives a member \(g\in\mathcal{D}\) with \(\|f-g\|_p<\varepsilon\). Thus \(\mathcal{D}\) is countable and dense, proving the theorem. \(\square\)

Worked Examples

A Common Pitfall: Approximating on the Whole Space

On an infinite-measure space, a small or even finite error on a bounded region does not imply that the global symmetric-difference measure is small. For example, on \((0,\infty)\) with Lebesgue measure, let \(E=(0,\infty)\) and \(B=(0,N)\). Then \(m(E\triangle B)=m([N,\infty))=\infty\), not a small finite error. A mistaken argument might try to approximate \(E\) globally by such bounded sets and then use \(\chi_B\) as a close \(L^p\) approximation to \(\chi_E\); in fact neither indicator belongs to \(L^p\) for finite \(p\).

The correction is to localize first. With \(F_N=(0,N)\), we have \(E\cap F_N=B\cap F_N=F_N\), so the restricted symmetric-difference error is zero. For a general \(L^p\) function, localization works because \(\int_{X\setminus F_N}|f|^p\,d\mu\to0\), as follows from the Monotone Convergence Theorem on the increasing sets \(F_N\). Only after this tail is controlled should one approximate sets and coefficients within a finite-measure region.

The argument separates the two sources of countability and approximation. Sigma-finiteness supplies finite-measure regions that control the \(L^p\) tails; countable generation modulo null sets supplies a countable algebra that approximates measurable sets inside those regions. Rational coefficients then turn the resulting simple functions into a countable dense family. If the functions are complex-valued, replacing \(\mathbb{Q}\) by the countable set \(\mathbb{Q}+i\mathbb{Q}\) gives the same conclusion.