Measuring Graphs and Diagonals by Sections

Why Sections Measure Graphs

A graph may look geometrically thin, but its product measure depends on the measures used on the two coordinate spaces. Under planar Lebesgue measure, the graph of a measurable real-valued function has measure zero. In a product with counting measure, however, a graph can have positive or infinite measure. Sections make these differences exact: they reduce the product measure of a set to integrals of measures of its one-dimensional slices.

The key tool is Tonelli’s theorem for nonnegative functions. We will use it only for sigma-finite measure spaces. This hypothesis is essential: the section formula cannot simply be applied to arbitrary measure spaces.

Sections and the Product-Measure Formula

Let \((X,\mathcal A)\) and \((Y,\mathcal B)\) be measurable spaces, and let \(E\subseteq X\times Y\). For \(x\in X\), its section in \(Y\) is \(E_x=\{y\in Y:(x,y)\in E\}\). For \(y\in Y\), its section in \(X\) is \(E^y=\{x\in X:(x,y)\in E\}\). If \(E\in\mathcal A\otimes\mathcal B\), the sections are measurable in their respective spaces.

Theorem (Tonelli’s section formula for sets): Suppose \(\mu\) on \((X,\mathcal A)\) and \(\nu\) on \((Y,\mathcal B)\) are sigma-finite. For every \(E\in\mathcal A\otimes\mathcal B\), the section-measure functions are measurable and

(\mu\otimes\nu)(E) =\int_X \nu(E_x)\,\mu(dx) =\int_Y \mu(E^y)\,\nu(dy).

The equalities are valid as equalities of extended nonnegative numbers, so the common value may be infinite.

This is Tonelli’s theorem applied to the indicator function \(1_E\). Sigma-finiteness is required for this version of the product-measure theorem. In particular, every use of the formula below has sigma-finite measures in both coordinates.

For a function \(f:X\to Y\), its graph is \(\operatorname{Gr}(f)=\{(x,f(x)):x\in X\}\). The vertical section at \(x\) is either the singleton \(\{f(x)\}\) or, for a graph restricted to part of its domain, the empty set. A horizontal section at \(y\) is the fiber \(f^{-1}(\{y\})\). These two types of sections can have very different measures.

Graphs in Two Lebesgue Coordinates

Take Lebesgue measure \(\lambda\) on \(\mathbb R\). It is sigma-finite, since \(\mathbb R=\bigcup_{n\in\mathbb N}[-n,n]\) and each interval in this union has finite measure. If \(f:\mathbb R\to\mathbb R\) is Lebesgue measurable, its graph is measurable in the product sigma-algebra \(\mathcal L\otimes\mathcal L\), where \(\mathcal L\) denotes the Lebesgue sigma-algebra. Indeed, the map \((x,y)\mapsto(f(x),y)\) is measurable from \((\mathbb R^2,\mathcal L\otimes\mathcal L)\) to the Borel plane, and the graph is its inverse image of the closed diagonal.

Theorem: If \(f:\mathbb R\to\mathbb R\) is Lebesgue measurable, then

(\lambda\otimes\lambda)(\operatorname{Gr}(f))=0.

Proof. For every \(x\in\mathbb R\), the vertical section of the graph is the singleton \(\{f(x)\}\). A singleton has Lebesgue measure zero. The section formula applies because both Lebesgue measures are sigma-finite; therefore

(\lambda\otimes\lambda)(\operatorname{Gr}(f)) =\int_{\mathbb R}\lambda(\{f(x)\})\,\lambda(dx) =\int_{\mathbb R}0\,\lambda(dx) =0.

\(\square\)

Worked Example: The Diagonal in the Lebesgue Plane

Let \(\Delta=\{(x,y)\in\mathbb R^2:y=x\}\). It is the graph of the measurable identity function \(f(x)=x\), so the theorem gives \((\lambda\otimes\lambda)(\Delta)=0\). Directly, \(\Delta_x=\{x\}\), and hence \(\lambda(\Delta_x)=0\) for every \(x\). In the other direction, \(\Delta^y=\{y\}\), which also has measure zero for every \(y\). Thus both section integrals are zero. The diagonal is unbounded and has infinitely many points, but neither fact prevents its two-dimensional Lebesgue measure from being zero.

Worked Example: The Graph of a Quadratic Function

Consider \(f(x)=x^2\). This is continuous, hence Lebesgue measurable, so its graph has product measure zero. The horizontal section at \(y\) can also be described explicitly. If \(y<0\), it is empty; if \(y=0\), it is \(\{0\}\); and if \(y>0\), it is \(\{-\sqrt y,\sqrt y\}\). Every such section is finite and therefore has Lebesgue measure zero. The two section calculations agree:

\int_{\mathbb R}\lambda(\{f(x)\})\,\lambda(dx)=0, \qquad \int_{\mathbb R}\lambda(\{x:x^2=y\})\,\lambda(dy)=0.

The horizontal description is special to this example: for other measurable functions a fiber can be infinite or even have positive Lebesgue measure. The vertical-section argument still proves that the graph has zero product measure.

When One Coordinate Uses Counting Measure

Let \(\mathbb N\) carry counting measure \(\#\), defined by \(\#(C)=|C|\) for finite \(C\) and \(\#(C)=\infty\) for infinite \(C\). This measure is sigma-finite because \(\mathbb N\) is a countable union of singletons. In contrast, counting measure on an uncountable set is not sigma-finite: any set of finite counting measure is finite, and countably many finite sets cannot cover an uncountable space.

For a measurable set \(A\subseteq\mathbb R\) and a measurable function \(f:A\to\mathbb N\), consider its graph in \(A\times\mathbb N\). The vertical section at \(x\in A\) contains exactly one point, and that point has counting measure one. Tonelli therefore yields the following formula.

Theorem: With Lebesgue measure restricted to \(A\) in the first coordinate and counting measure on \(\mathbb N\) in the second, the measurable graph of \(f:A\to\mathbb N\) satisfies

(\lambda|_A\otimes\#)(\operatorname{Gr}(f))=\lambda(A).

Proof. The restricted Lebesgue measure is sigma-finite, and counting measure on \(\mathbb N\) is sigma-finite. For each \(x\in A\), the graph section is \(\{f(x)\}\), of counting measure one. Thus the section formula gives

(\lambda|_A\otimes\#)(\operatorname{Gr}(f)) =\int_A \#(\{f(x)\})\,\lambda(dx) =\int_A 1\,\lambda(dx) =\lambda(A).

This includes the case \(\lambda(A)=\infty\). \(\square\)

Worked Example: A Graph with Positive Mixed Product Measure

Let \(A=[0,3)\), and define \(f:A\to\mathbb N\) by \(f(x)=1\) for \(0\le x<1\) and \(f(x)=2\) for \(1\le x<3\). This function is measurable. Its graph is the disjoint union \(([0,1)\times\{1\})\cup([1,3)\times\{2\})\). The rectangle measures are \(1\cdot1=1\) and \(2\cdot1=2\), respectively, so the graph measure is \(3\). Equivalently, each vertical section has counting measure one and the domain has Lebesgue measure three. Thus a graph that has zero measure in a Lebesgue–Lebesgue product has positive measure in this mixed product.

Reversing the roles of the coordinates gives a different conclusion. If \(f:\mathbb N\to\mathbb R\) is any function, its graph is countable and is measurable in the product of counting measure and Lebesgue measure. Each vertical section in the real coordinate is either empty or a singleton, so the graph has product measure zero. The horizontal section at \(y\) is the fiber \(\{n:f(n)=y\}\), which may contain many integers. Nevertheless it is nonempty only for \(y\) in the countable range \(f(\mathbb N)\), a Lebesgue-null set. Tonelli gives zero from either order.

Worked Example: A Countable-Domain Graph Has Zero Mixed Measure

Let \(f:\mathbb N\to\mathbb R\) be \(f(n)=n\). In \(\mathbb N\times\mathbb R\), the graph is the diagonal \(\{(n,n):n\in\mathbb N\}\). For each \(n\), its section in \(\mathbb R\) is \(\{n\}\), so its Lebesgue measure is zero. Since counting measure on \(\mathbb N\) and Lebesgue measure on \(\mathbb R\) are sigma-finite, Tonelli applies and gives

(\#\otimes\lambda)(\operatorname{Gr}(f)) =\sum_{n\in\mathbb N}\lambda(\{n\}) =\sum_{n\in\mathbb N}0 =0.

In the reverse section calculation, the fiber at \(y\) is \(\{y\}\) when \(y\in\mathbb N\), and empty otherwise. Its counting measure is one on \(\mathbb N\) and zero elsewhere, so its integral against Lebesgue measure is also zero.

Counting Measure in Both Coordinates

Counting measure on \(\mathbb N\) in each coordinate is again sigma-finite. If \(f:\mathbb N\to\mathbb N\), the graph has one point for each input, and each point has product counting measure one. Consequently the graph measure is \(\infty\). More generally, restricting the domain to a finite set of \(m\) integers gives graph measure \(m\). For the diagonal \(\{(n,n):n\in\mathbb N\}\), this is the countably infinite value, not zero.

This computation contrasts with the Lebesgue diagonal: its sections are singletons in both settings, but singleton measure is zero for Lebesgue measure and one for counting measure. A section argument is only as informative as the measures assigned to the sections.

Worked Example: The Diagonal for Counting Measure

Let \(\Delta_{\mathbb N}=\{(n,n):n\in\mathbb N\}\). Its section at each \(n\) is \(\{n\}\), which has counting measure one. The set-section formula gives

(\#\otimes\#)(\Delta_{\mathbb N}) =\sum_{n\in\mathbb N}\#(\{n\}) =\sum_{n\in\mathbb N}1 =\infty.

Equivalently, each horizontal section also has counting measure one, so integrating those section measures over \(\mathbb N\) gives the same infinite value.

How to Avoid a Section-Measure Pitfall

When applying sections to a graph, make three checks. First, establish that the set is measurable in the product sigma-algebra. Second, verify that both coordinate measures are sigma-finite before invoking the section formula. Third, compute the measure of a section under the measure assigned to its own coordinate; do not treat every singleton as null.

The sigma-finiteness check is not a formality. If counting measure is placed on an uncountable set, it is not sigma-finite, so the Tonelli theorem stated here does not apply. There are non-sigma-finite product-measure examples for which the section integrals do not equal the product measure. One must not infer a product-measure value merely by writing down iterated section integrals in that setting.

The same caution applies to infinite section measures. Tonelli’s theorem handles nonnegative section measures, including infinite values, and does not require their integral to be finite. Fubini’s theorem for signed integrands has different integrability requirements; it is not needed for these graph calculations.

Takeaway: Under sigma-finite measures, a measurable set’s product measure is the integral of its section measures. For real-valued measurable graphs, Lebesgue singletons make the graph null in the Lebesgue plane; counting singletons instead have mass one, so the product measure can change substantially when a coordinate measure changes.

Check Your Understanding

Use the section formula and check its hypotheses before computing.

Why is sigma-finiteness relevant when applying Tonelli to a measurable graph?
What is the product Lebesgue measure of the graph of a measurable function from \(\mathbb R\) to \(\mathbb R\), and which sections establish the result?
For a measurable function \(f:[0,5]\to\mathbb N\), what is the measure of its graph under Lebesgue measure on the domain and counting measure on \(\mathbb N\)?
Why does the graph of \(f:\mathbb N\to\mathbb R\) have zero counting–Lebesgue product measure?
What is the counting–counting measure of the diagonal in \(\mathbb N\times\mathbb N\)?
Why does the sigma-finite section formula not directly apply to counting measure on an uncountable set?