Measurability of Maximal Functions

Why Measurability Needs Attention

The centered Hardy–Littlewood maximal function is defined by taking a supremum over every positive radius, and the uncentered version takes a supremum over every ball containing the point. A supremum over an uncountable family of measurable functions is not automatically measurable. The remedy is to show that ball averages vary continuously with their parameters, then replace the uncountable choices by countable dense families.

Throughout, we work on \((\mathbb{R}^n,\mathcal{L},m)\), with Lebesgue measure, which is complete and sigma-finite. Let \(f:\mathbb{R}^n\to\mathbb{R}\) or \(\mathbb{C}\) be Lebesgue measurable and locally integrable, meaning that \(\int_K |f|\,dm<\infty\) for every compact set \(K\). The maximal functions below are nonnegative extended-real-valued functions; they may take the value \(+\infty\). Write \(B(x,r)=\{y:|y-x|<r\}\), and let \(v_n=m(B(0,1))\), so \(m(B(x,r))=v_nr^n>0\).

Each average is finite: the closure of any ball is compact, and local integrability makes the integral of \(|f|\) over that closure finite. We will show that both maximal functions are Lebesgue measurable as extended-real-valued functions.

Continuity of Ball Integrals

The basic fact is that changing the center or radius slightly changes a ball only by a thin shell. Here is the precise measure estimate behind that statement. Suppose \(c_j\to c\) in \(\mathbb{R}^n\) and \(r_j\to r>0\), and put \(\delta_j=|c_j-c|+|r_j-r|\). If a point \(y\) belongs to exactly one of \(B(c_j,r_j)\) and \(B(c,r)\), the triangle inequality gives \(||y-c|-r|\le\delta_j\). Thus the symmetric difference lies in the shell around the sphere of radius \(r\) and thickness \(\delta_j\). For \(\delta_j<r\), the shell has measure at most \(v_n((r+\delta_j)^n-(r-\delta_j)^n)\), which tends to zero. This uses the volume formula for balls; the same estimate also shows that a sphere has measure zero.

Proof. Let \(c_j\to c\) and \(r_j\to r>0\). All the balls \(B(c_j,r_j)\) and \(B(c,r)\), apart from at most finitely many initial terms, lie in a common compact set \(K\). Denote their symmetric difference by \(E_j\). The shell estimate above gives \(m(E_j)\to0\).

We check directly that the integral of \(|f|\) over these differences tends to zero. Given \(\varepsilon>0\), choose \(T>0\) so that \(\int_{K\cap\{|f|>T\}}|f|\,dm<\varepsilon/2\). Such a \(T\) exists because \(|f|\) is integrable on \(K\). For any measurable \(E\subset K\), \[ \int_E|f|\,dm\le Tm(E)+\int_{K\cap\{|f|>T\}}|f|\,dm. \] For all sufficiently large \(j\), \(Tm(E_j)<\varepsilon/2\); hence \(\int_{E_j}|f|\,dm<\varepsilon\). The difference between the two ball integrals is bounded in absolute value by this quantity, so the integrals converge. Finally \(m(B(c_j,r_j))=v_nr_j^n\to v_nr^n=m(B(c,r))>0\), so division by volume preserves continuity. Fixing either the center or the radius gives the two stated continuity conclusions. \(\square\)

Centered Maximal Functions: Countable Radii

For fixed \(x\), continuity in the radius allows us to use positive rational radii without changing the supremum. Indeed, for any \(r>0\), choose positive rationals \(q_j\to r\). Then \(A_{q_j}f(x)\to A_rf(x)\). It follows that every value in the supremum over all radii is a limit of values at rational radii, so the two suprema agree.

Proof. For each positive rational \(q\), the function \(x\mapsto A_qf(x)\) is continuous by the proposition, and is therefore Lebesgue measurable. The countable-radius reduction gives \(M_cf(x)=\sup_{q\in\mathbb{Q},\,q>0}A_qf(x)\). For every real \(\alpha\), the strict superlevel set is \[ \{x:M_cf(x)>\alpha\}=\bigcup_{\substack{q\in\mathbb{Q}\\q>0}}\{x:A_qf(x)>\alpha\}. \] Each set on the right is open, since \(A_qf\) is continuous; the union is countable and hence open. In particular, these superlevel sets are Lebesgue measurable, which proves measurability of the extended-real-valued function \(M_cf\). (For \(\alpha<0\), the set is all of \(\mathbb{R}^n\), consistent with the same conclusion.) \(\square\)

Uncentered Maximal Functions: Countable Balls

For the uncentered function, use balls whose centers have rational coordinates and whose radii are positive rationals. These form a countable family. If \(x\in B(c,r)\), the inclusion is strict: \(r-|x-c|>0\). Consequently, sufficiently small perturbations of \(c\) and \(r\) still give a ball containing \(x\). Continuity of the ball integral, including its dependence on both parameters, ensures that the average also changes continuously.

Proof. Let \(\mathcal{Q}\) be the countable collection of balls \(B(q,s)\) with \(q\in\mathbb{Q}^n\) and \(s\in\mathbb{Q}\), \(s>0\). We first verify that \[ M_uf(x)=\sup_{\substack{B\in\mathcal{Q}\\x\in B}}\frac{1}{m(B)}\int_B|f|\,dm. \] The supremum over \(\mathcal{Q}\) cannot exceed the defining supremum, since every ball in \(\mathcal{Q}\) is an open ball containing \(x\). Conversely, take any open ball \(B(c,r)\) containing \(x\), and write its average as \(a\). Choose rational centers \(q_j\to c\) and positive rational radii \(s_j\to r\). Since \(r-|x-c|>0\), for all sufficiently large \(j\), \(s_j>|x-q_j|\), so \(x\in B(q_j,s_j)\). The proposition gives convergence of the integrals, and the volumes converge to \(v_nr^n>0\); therefore the averages over \(B(q_j,s_j)\) converge to \(a\). Taking the supremum over the countable family thus gives at least every admissible average \(a\), proving the equality.

For a real \(\alpha\), that equality yields \[ \{x:M_uf(x)>\alpha\} = \bigcup_{\substack{B\in\mathcal{Q}\\m(B)^{-1}\int_B|f|\,dm>\alpha}} B. \] Indeed, a supremum is strictly greater than \(\alpha\) exactly when at least one member of the family is strictly greater than \(\alpha\). The union is countable and consists of open balls, so it is open. This proves that \(M_uf\) is Lebesgue measurable. \(\square\)

A Simple Check and a Common Pitfall

Common mistake: restricting to integer radii. Countability alone does not justify replacing all radii by the integers; the smaller countable family may miss the supremum. For the triangular function in the earlier example, the average at \(0\) is \(1-r/2\) for \(0<r\le1\), and \(1/(2r)\) for \(r\ge1\). At every integer radius \(n\ge1\), it is \(1/(2n)\), whose supremum is \(1/2\). But the supremum over all positive radii is \(1\), approached as \(r\downarrow0\). Positive rational radii work because they are dense and the average is continuous in \(r\), not merely because they are countable.

The same distinction explains why the continuity argument is essential for the centered function, while the uncentered proof also needs to preserve the condition that the ball contains \(x\). Strict containment in an open ball gives room to perturb its rational center and radius. With these two ingredients, both uncountable definitions reduce to countable descriptions of their superlevel sets, and measurability follows without assumptions that the maximal functions are finite.