What Separability Means in \(L^p(\mathbb{R}^n)\)
A metric space is separable if it contains a countable dense subset. For \(L^p\), density means approximation in the norm, not necessarily pointwise approximation. The key is to reduce arbitrary functions to finite simple functions, then replace their measurable level sets by finite unions of boxes with rational endpoints and their coefficients by rational numbers.
Throughout, \(n\ge1\), \(1\le p<\infty\), and the measure space is \((\mathbb{R}^n,\mathcal{L},m)\), with Lebesgue measure. This space is complete and sigma-finite, but not finite. Functions are real-valued and measurable, with functions equal \(m\)-a.e. identified in \(L^p(\mathbb{R}^n)\). The norm is \(\|f\|_p=(\int_{\mathbb{R}^n}|f|^p\,dm)^{1/p}\), and \(L^p\) consists of classes with finite norm.
For rational numbers \(a_i<b_i\), a bounded half-open box is \(Q=\prod_{i=1}^n[a_i,b_i)\). Let \(\mathcal{Q}\) be the family of all such boxes, and define
Each box has finite measure, so its indicator belongs to \(L^p\). The same holds for every finite sum in \(\mathcal{D}\). The family \(\mathcal{Q}\) is countable because each box is specified by a finite list of rational endpoints. For every fixed \(k\), the possible \(k\)-term sums form a countable set: they are specified by \(k\) rational coefficients and \(k\) boxes from a countable family. Taking the union over the countably many values of \(k\) shows that \(\mathcal{D}\) is countable. Countability alone does not show density; we next establish the needed approximation of sets.
Approximating Measurable Sets by Rational Boxes
We use regularity of Lebesgue measure: if \(E\in\mathcal{L}\) has finite measure and \(\eta>0\), there is an open set \(U\supseteq E\) with \(m(U\setminus E)<\eta\). This is the standard regularity result for Lebesgue measure. The following argument turns it into approximation by finite unions of rational boxes.
Proof. Choose an open \(U\supseteq E\) with \(m(U\setminus E)<\varepsilon/2\). Since \(m(U)=m(E)+m(U\setminus E)<\infty\), \(U\) has finite measure. Every point of \(U\) lies in a rational-endpoint half-open box contained in \(U\): openness gives a small neighborhood within \(U\), and rational endpoints can be chosen around the point inside that neighborhood. Thus \(U\) is the union of the countable family of boxes in \(\mathcal{Q}\) that are contained in \(U\). Enumerate them as \(Q_1,Q_2,\ldots\), allowing a finite enumeration if there are only finitely many, and put \(F_N=\bigcup_{j=1}^NQ_j\). The sets \(F_N\) increase to \(U\). Continuity from below of measure gives \(m(F_N)\to m(U)\), hence \(m(U\setminus F_N)\to0\). Choose \(N\) with \(m(U\setminus F_N)<\varepsilon/2\), and set \(F=F_N\). Since \(E\subseteq U\) and \(F\subseteq U\), every point of \(E\triangle F\) lies either in \(U\setminus E\) or in \(U\setminus F\). Therefore $$ m(E\triangle F)\le m(U\setminus E)+m(U\setminus F)<\varepsilon. $$ This proves the lemma. \(\square\)
A finite union \(F\) of rational boxes need not itself be a single box, but its indicator belongs to the rational span of box indicators. Indeed, inclusion-exclusion expresses the indicator of a finite union as an alternating sum of indicators of intersections. An intersection of finitely many boxes in \(\mathcal{Q}\) is either empty or another box in \(\mathcal{Q}\). This observation lets us use such \(F\) without enlarging the proposed dense set.
Worked Example: Approximating an Interval with an Irrational Endpoint
On \((\mathbb{R},\mathcal{L},m)\), let \(E=[\sqrt{2},\sqrt{2}+1)\) and \(F=[7/5,5/2)\). Both sets have finite measure, and \(F\) is a rational-endpoint box. Since \(7/5<\sqrt{2}<\sqrt{2}+1<5/2\), the symmetric difference consists of the two end pieces. Their lengths are \(\sqrt{2}-7/5\) and \(5/2-(\sqrt{2}+1)\), so
For every \(1\le p<\infty\), the difference of the indicators has absolute value \(1\) on \(E\triangle F\) and \(0\) elsewhere. Consequently, $$ \|\chi_E-\chi_F\|_p =\left(\int_{\mathbb{R}}|\chi_E-\chi_F|^p\,dm\right)^{1/p} =\left(\frac{1}{10}\right)^{1/p}. $$ Check: for \(p=1\) the norm is \(1/10\); for \(p=2\) it is \(1/\sqrt{10}\). In each case, raising the norm to the power \(p\) recovers the symmetric-difference measure \(1/10\).
The Density Theorem
We use two standard ingredients from the earlier material on \(L^p\) density and inequalities. First, finite simple functions supported on finite-measure measurable sets are dense in \(L^p\): given \(f\in L^p\) and \(\rho>0\), there is \(s=\sum_{j=1}^k c_j\chi_{E_j}\), with real \(c_j\) and disjoint \(E_j\in\mathcal{L}\) of finite measure, such that \(\|f-s\|_p<\rho\). Second, Minkowski’s inequality gives \(\|\sum_j h_j\|_p\le\sum_j\|h_j\|_p\). We now prove the approximation specific to \(\mathcal{D}\).
Proof. Fix \(f\in L^p(\mathbb{R}^n)\) and \(\varepsilon>0\). By density of finite simple functions, choose \(s=\sum_{j=1}^k c_j\chi_{E_j}\) with \(\|f-s\|_p<\varepsilon/2\), where the \(E_j\) have finite measure. If \(k=0\), then \(s=0\in\mathcal{D}\) and the desired approximation is already obtained. Suppose \(k\ge1\).
For each \(j\), choose a finite union \(F_j\) of rational boxes so that \(m(E_j\triangle F_j)\) is as small as needed. The indicator identity gives $$ \|\chi_{E_j}-\chi_{F_j}\|_p=m(E_j\triangle F_j)^{1/p}, \qquad \|\chi_{F_j}\|_p=m(F_j)^{1/p}. $$ The first equality follows by integrating the \(p\)-th power of the indicator of the symmetric difference. By the lemma, we can make the first quantity arbitrarily small. We may require also \(m(E_j\triangle F_j)<1\), so \(m(F_j)\le m(E_j)+1\), giving a finite bound for the second quantity.
Choose \(F_j\) so that \(|c_j|\|\chi_{E_j}-\chi_{F_j}\|_p<\varepsilon/(4k)\) for every \(j\) with \(c_j\ne0\); if \(c_j=0\), this term is zero without a restriction. Because \(\mathbb{Q}\) is dense in \(\mathbb{R}\), choose \(q_j\in\mathbb{Q}\) with $$ |c_j-q_j|\,m(F_j)^{1/p}<\frac{\varepsilon}{4k}. $$ If \(m(F_j)=0\), this inequality holds for any rational \(q_j\), and the corresponding indicator is zero in \(L^p\). Define \(g=\sum_{j=1}^kq_j\chi_{F_j}\). By inclusion-exclusion, \(g\) is a finite rational linear combination of indicators of boxes in \(\mathcal{Q}\), so \(g\in\mathcal{D}\).
For each \(j\), write \(c_j\chi_{E_j}-q_j\chi_{F_j}=c_j(\chi_{E_j}-\chi_{F_j})+(c_j-q_j)\chi_{F_j}\). Minkowski’s inequality and the choices above give $$ \|s-g\|_p \le\sum_{j=1}^k\left(|c_j|\|\chi_{E_j}-\chi_{F_j}\|_p+ |c_j-q_j|\|\chi_{F_j}\|_p\right) <\frac{\varepsilon}{2}. $$ The triangle inequality now yields \(\|f-g\|_p\le\|f-s\|_p+\|s-g\|_p<\varepsilon\). Thus every \(L^p\) element is arbitrarily close to an element of the countable set \(\mathcal{D}\), proving density and separability. \(\square\)
Worked Example: Replacing a Real Coefficient by a Rational One
On \((\mathbb{R},\mathcal{L},m)\), take \(f=\sqrt{2}\,\chi_{[0,1)}\) and \(g=(7/5)\chi_{[0,1)}\). The function \(g\) belongs to \(\mathcal{D}\). Since \(\sqrt{2}>7/5\), direct integration gives, for every \(1\le p<\infty\),
Check: the interval has measure \(1\), so the integral of the constant error to the \(p\)-th power is exactly \((\sqrt{2}-7/5)^p\); taking its \(p\)-th root gives the displayed norm. Choosing a rational number closer to \(\sqrt{2}\) makes this error arbitrarily small.
Worked Example: Approximating \(x\) by Rational Step Values
On \((\mathbb{R},\mathcal{L},m)\), let \(f(x)=x\) for \(0\le x<1\) and \(f(x)=0\) elsewhere. For \(k=0,1,2,3\), define \(Q_k=[k/4,(k+1)/4)\) and set \(g=\sum_{k=0}^3(k/4)\chi_{Q_k}\). This is an element of \(\mathcal{D}\). On \(Q_k\), \(f(x)-g(x)=x-k/4\), which ranges from \(0\) up to \(1/4\). Therefore, for \(1\le p<\infty\),
Hence \(\|f-g\|_p=\frac14(p+1)^{-1/p}\). Check: at \(p=1\), the error is \(1/8\). Independently, on each of the four intervals the error is a triangle of base \(1/4\) and height \(1/4\), with area \(\frac12(1/4)(1/4)=1/32\); the total is \(4/32=1/8\), as calculated.
A Common Pitfall
Having rational coefficients does not by itself make a family countable if the boxes are allowed arbitrary real endpoints. For example, the intervals \([a,a+1)\), \(a\in\mathbb{R}\), form an uncountable family. If \(a\ne b\), these intervals differ on a set of positive measure, so their indicators represent distinct \(L^p\) elements. The correction is to restrict every endpoint to \(\mathbb{Q}\); the resulting box family is countable while still rich enough for approximation in measure.
A second error is to identify small set error with small \(L^p\) error without taking the correct power. For indicators the exact relation is \(\|\chi_E-\chi_F\|_p=m(E\triangle F)^{1/p}\), not \(m(E\triangle F)\) in general. This is why approximation in symmetric-difference measure supplies norm approximation for every finite \(p\), while the numerical norm depends on \(p\).
Check Your Understanding
Use the set-approximation lemma and the norm estimates to justify each answer.
- Why is the family of bounded boxes with rational endpoints countable in \(\mathbb{R}^n\)?
- If \(m(E\triangle F)=1/16\), what is \(\|\chi_E-\chi_F\|_p\) for \(p=1\) and for \(p=2\)?
- Explain why a finite union of rational boxes has an indicator in the rational span of indicators of rational boxes.
- Which part of the proof uses \(p<\infty\), and why does the indicator approximation estimate have the form \(m(E\triangle F)^{1/p}\)?
- Give an example of an uncountable family that results if arbitrary real endpoints are allowed, and explain why restricting endpoints to rationals fixes the countability issue.