Continuous Function Density for Regular Borel Measures

From Regularity to Continuous Approximation

In \(L^p\), functions that agree almost everywhere represent the same element. This makes approximation of indicators a natural starting point: if a finite-measure Borel set can be enclosed by an open set and contains a compact set with small measure error, a continuous function can be built to equal one on the compact part and vanish outside the open part. Its discrepancy from the indicator is then confined to sets of small measure. The argument applies far beyond Euclidean space, provided the topology and measure have the right regularity.

Throughout, \(X\) is a locally compact, second-countable Hausdorff space, \(\mathcal{B}_X\) is its Borel sigma-algebra, and \(\mu\) is a sigma-finite Radon measure on \((X,\mathcal{B}_X)\). Here sigma-finite means \(X\) is a countable union of measurable sets of finite measure; it does not mean \(\mu(X)<\infty\). A Radon measure is finite on compact sets and regular: Borel sets can be approximated from outside by open sets, and open sets from inside by compact sets. Functions in \(L^p(\mu)\) are Borel-measurable real- or complex-valued functions, identified when they agree \(\mu\)-a.e. We consider \(1\le p<\infty\), and use the norm \(\|f\|_p=(\int_X|f|^p\,d\mu)^{1/p}\). The space \(C_c(X)\) consists of continuous functions with compact support, over the same scalar field.

The topological ingredient is the standard compact-support Urysohn lemma for locally compact Hausdorff spaces: if \(K\subset U\), where \(K\) is compact and \(U\) is open, there is a function \(h\in C_c(X)\) with \(0\le h\le1\), \(h=1\) on \(K\), and \(\operatorname{supp}(h)\subset U\). This is a standard result of general topology. The measure-theoretic argument below uses precisely this form.

Approximating a Finite-Measure Set

Proof. By inner regularity, choose a compact set \(K\subset E\) with \(\mu(E\setminus K)<\eta/2\). By outer regularity, choose an open set \(U\supset E\) with \(\mu(U\setminus E)<\eta/2\). In particular, \(K\subset U\). The compact-support Urysohn lemma gives \(h\in C_c(X)\) with \(0\le h\le1\), \(h=1\) on \(K\), and support contained in \(U\). On \(K\), \(h=\chi_E=1\). On \(X\setminus(E\cup U)\), both functions are zero. Everywhere else their absolute difference is at most one, so \[ |\chi_E-h|^p\le \chi_{E\setminus K}+\chi_{U\setminus E}. \] Integrating and using the two choices of approximating sets gives \[ \|\chi_E-h\|_p^p \le \mu(E\setminus K)+\mu(U\setminus E) <\eta. \] This proves the proposition. \(\square\)

The estimate explains the role of both approximations. The compact set ensures the cutoff equals the indicator on most of \(E\); the open set confines the cutoff’s possible extra values to a small-measure region outside \(E\). The measure of \(E\) is finite so that \(\chi_E\in L^p(\mu)\). No assumption that \(\mu(X)\) is finite is needed.

Extending from Indicators to All of \(L^p\)

We use the simple-function density result for finite-\(p\) spaces: for \(f\in L^p(\mu)\) and any positive tolerance, there is a simple function \(s=\sum_{j=1}^N c_j\chi_{E_j}\) with \(\mu(E_j)<\infty\) and \(\|f-s\|_p\) smaller than that tolerance. The sets can be taken Borel here. In the completed measure space, a completed-measurable function has a Borel representative up to a null set, so the same conclusion holds for its \(L^p\) class. This simple-function density is the standard reduction from the earlier material on simple-function approximation; the new step is replacing each indicator by a continuous function.

Proof of the theorem. Fix \(f\in L^p(\mu)\) and \(\varepsilon>0\). Choose a finite-measure-supported simple function \(s=\sum_{j=1}^N c_j\chi_{E_j}\) such that \(\|f-s\|_p<\varepsilon/2\). If all \(c_j=0\), then \(s=0\) and \(g=0\) already gives \(\|f-g\|_p<\varepsilon\). Otherwise, set \(C=\sum_{j=1}^N|c_j|>0\). For each \(j\) with \(c_j\ne0\), the proposition supplies \(h_j\in C_c(X)\) with \(0\le h_j\le1\) and \[ \|\chi_{E_j}-h_j\|_p<\frac{\varepsilon}{2C}. \] For \(c_j=0\), set \(h_j=0\). Define \(g=\sum_{j=1}^N c_jh_j\). A finite sum of continuous functions is continuous, and a finite union of compact supports is compact, so \(g\in C_c(X)\). By the triangle inequality in \(L^p\), \[ \|s-g\|_p \le\sum_{j=1}^N|c_j|\,\|\chi_{E_j}-h_j\|_p <\sum_{j:c_j\ne0}|c_j|\frac{\varepsilon}{2C} =\frac{\varepsilon}{2}. \] Applying the triangle inequality once more yields \[ \|f-g\|_p\le\|f-s\|_p+\|s-g\|_p<\frac{\varepsilon}{2}+\frac{\varepsilon}{2} =\varepsilon. \] Thus \(C_c(X)\) is dense in \(L^p(\mu)\). \(\square\)

The proof separates the roles of the assumptions. Sigma-finiteness permits reduction to simple functions supported on finite-measure sets. Radon regularity supplies the compact and open approximations. Local compactness and the Hausdorff condition allow a compactly supported continuous cutoff between them. The finite-\(p\) restriction matters: the argument controls the \(p\)-th power of the error by the measure of the disagreement region.

A Common Pitfall: An Indicator Is Not a Cutoff

For Lebesgue measure on \(\mathbb{R}\), the indicator \(\chi_{[0,1]}\) is not continuous, so it is not itself an element of \(C_c(\mathbb{R})\). A common mistake is to call it a continuous approximation merely because it has compact support. Compact support does not imply continuity. The ramp example gives the correction: the approximants are continuous, agree with the indicator except on transition intervals, and their \(L^p\) error is \(2\delta/(p+1)\), which can be made smaller than any prescribed positive number.

Another important distinction is that sigma-finiteness does not make every indicator eligible for this argument. If \(\mu(E)=\infty\), then \(\chi_E\notin L^p(\mu)\) for finite \(p\), since \(\int|\chi_E|^p\,d\mu=\mu(E)=\infty\). The density proof instead approximates \(L^p\) functions by simple functions whose level sets have finite measure. No claim is being made that an infinite-measure indicator is approximable in \(L^p\).

Why the Result Is Useful

The theorem lets one test or construct \(L^p\) functions using continuous functions with controlled support, even when the underlying measure is weighted or the space is not Euclidean. The key quantitative statement is the set estimate: an error bounded by one and confined to a set of measure less than \(\eta\) has \(p\)-th power of its \(L^p\) norm less than \(\eta\). Regularity supplies precisely that small region of disagreement. The result does not say every \(L^p\) function has a continuous representative; it says continuous compactly supported functions are dense in the norm.