Why Sections Matter
A set in a product space can be examined one coordinate at a time. Fixing the first coordinate produces a section in the second space; fixing the second produces a section in the first. Sections let us describe the slices of a set and, later, study functions by examining their level sets one slice at a time.
The central fact is a measurability guarantee: if a set belongs to the product sigma-algebra, each of its sections is measurable in the relevant factor space. The result requires no probability measure, and it holds for arbitrary measurable spaces. Applying it to the level sets of a function gives the corresponding conclusion for function sections.
Sections of Sets
Let \((X,\Sigma)\) and \((Y,\mathcal{T})\) be measurable spaces. Their product sigma-algebra, denoted \(\Sigma\otimes\mathcal{T}\), is the sigma-algebra on \(X\times Y\) generated by measurable rectangles \(A\times B\), where \(A\in\Sigma\) and \(B\in\mathcal{T}\).
The subscript fixes the \(X\)-coordinate and leaves a set in \(Y\); the superscript fixes the \(Y\)-coordinate and leaves a set in \(X\). The two orientations are worth keeping distinct because their measurability is asserted in different sigma-algebras.
Every measurable rectangle belongs to \(\mathcal{C}_x\). In fact, for \(A\in\Sigma\) and \(B\in\mathcal{T}\), \((A\times B)_x=B\) if \(x\in A\), and \((A\times B)_x=\varnothing\) if \(x\notin A\). Both possible sections belong to \(\mathcal{T}\). Since \(\mathcal{C}_x\) is a sigma-algebra containing all measurable rectangles, it contains \(\Sigma\otimes\mathcal{T}\). Thus \(E_x\in\mathcal{T}\) for every \(E\in\Sigma\otimes\mathcal{T}\). Applying the same argument with the roles of \(X\) and \(Y\) interchanged gives \(E^y\in\Sigma\) for every \(y\in Y\).
The proof uses the defining construction of the product sigma-algebra: it checks the claim on rectangles, then observes that the sets satisfying the claim form a sigma-algebra. The result is pointwise in the fixed coordinate. There is no exceptional set of coordinates.
Worked Example: Sections of a Union of Rectangles
Let \(A,C\in\Sigma\) and \(B,D\in\mathcal{T}\), and set \(E=(A\times B)\cup(C\times D)\). This is product-measurable because it is a finite union of measurable rectangles. Fix \(x\in X\). If \(x\in A\) and \(x\in C\), then \(E_x=B\cup D\). If \(x\in A\) and \(x\notin C\), then \(E_x=B\). If \(x\notin A\) and \(x\in C\), then \(E_x=D\). If \(x\notin A\) and \(x\notin C\), then \(E_x=\varnothing\). Each case gives a member of \(\mathcal{T}\).
For a fixed \(y\in Y\), the same membership test gives \(E^y=A\cup C\) if \(y\in B\) and \(y\in D\), \(E^y=A\) if \(y\in B\) and \(y\notin D\), \(E^y=C\) if \(y\notin B\) and \(y\in D\), and \(E^y=\varnothing\) if \(y\notin B\) and \(y\notin D\). Each belongs to \(\Sigma\), as the theorem predicts.
Sections of Measurable Functions
For a function on \(X\times Y\), fixing one coordinate produces a function on the other space. If \(f:X\times Y\to[0,\infty]\), define \(f_x:Y\to[0,\infty]\) by \(f_x(y)=f(x,y)\). For \(f:X\times Y\to\mathbb{R}\), define \(f_x\) in the same way, with values in \(\mathbb{R}\). The other orientation is \(f^y(x)=f(x,y)\).
A useful measurability criterion is that a nonnegative extended-real-valued function is measurable if and only if its strict upper level sets \(\{z:f(z)>a\}\) are measurable for every \(a\in\mathbb{R}\). For real-valued functions the same criterion applies. These rays determine measurability because they generate the Borel sigma-algebra: open intervals can be formed using countable unions and intersections of upper and lower rays. We use this standard criterion to transfer measurability from a function's level sets to its sections.
Now suppose \(f:X\times Y\to\mathbb{R}\) is measurable. The same construction gives measurable strict upper level sets for each section, since \(L_a=\{(u,v):f(u,v)>a\}\) is product-measurable for every \(a\in\mathbb{R}\). The real-valued version of the criterion therefore gives measurable sections in both orientations.
The level-set identity is the essential step: fixing a coordinate commutes with taking the set where a function exceeds a threshold. No integration is involved. This result establishes measurability of the sections themselves, a prerequisite for later questions about integrating functions along sections.
Worked Example: Sections of the Diagonal in the Plane
Take \(X=Y=\mathbb{R}\) with their Borel sigma-algebras, and let \(\Delta=\{(x,y)\in\mathbb{R}^2:x=y\}\). The diagonal is closed, since it is the inverse image of the closed set \(\{0\}\) under the continuous map \((x,y)\mapsto x-y\). It is therefore Borel in \(\mathbb{R}^2\). The Borel sigma-algebra on \(\mathbb{R}^2\) equals \(\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\): open sets are countable unions of open rectangles from a countable basis, and open rectangles are product-measurable.
For any fixed \(x\in\mathbb{R}\), \(\Delta_x=\{x\}\), a Borel singleton. For any fixed \(y\in\mathbb{R}\), \(\Delta^y=\{y\}\), also Borel. Thus the theorem applies to both types of section, and in this example each slice consists of exactly one point.
Worked Example: Sections of a Threshold Function
On \(\mathbb{R}^2\), define \(f(x,y)=\mathbf{1}_{\{(u,v):u+v>0\}}(x,y)\), where \(\mathbf{1}_E\) is \(1\) on \(E\) and \(0\) outside \(E\). The set \(\{(u,v):u+v>0\}\) is open because addition is continuous, so it is product-Borel. Its indicator \(f\) is therefore a measurable, nonnegative real-valued function.
Fix \(x\in\mathbb{R}\). Then \(f_x(y)=1\) exactly when \(x+y>0\), or equivalently \(y>-x\). For a real threshold \(a\), the upper level set is \(\{y:f_x(y)>a\}=\mathbb{R}\) when \(a<0\), \((-x,\infty)\) when \(0\leq a<1\), and \(\varnothing\) when \(a\geq1\). These sets are Borel in every case. For example, at \(a=0\) the strict inequality \(f_x(y)>0\) selects precisely the half-line \((-x,\infty)\); at \(a=1\) no value of \(f_x\) exceeds the threshold. This calculation verifies measurability directly by the level-set criterion.
Scope and a Common Pitfall
The conclusion is one-way: product measurability implies measurable sections. One should not use it as though it asserted that a set with measurable sections must belong to the product sigma-algebra. The proof establishes a property shared by every product-measurable set; it does not characterize the entire product sigma-algebra by that property.
It is also important not to confuse the two section orientations. For \(E\subseteq X\times Y\), \(E_x\) is a subset of \(Y\), so its measurability is tested against \(\mathcal{T}\). In contrast, \(E^y\) is a subset of \(X\), and its measurability is tested against \(\Sigma\). For functions, fixing the first coordinate likewise leaves a function on \(Y\), not \(X\).
The practical method is compact: for sets, check sections on rectangles and use sigma-algebra closure; for functions, take an arbitrary strict upper level set and identify the section of that set. This separates the measurability argument from any later analysis of integrals.
Check Your Understanding
Answer these questions using the definitions and arguments above.
- For a set \(E\subseteq X\times Y\), which factor sigma-algebra is used to test whether \(E_x\) is measurable?
- Why does checking the section property on measurable rectangles suffice for all sets in \(\Sigma\otimes\mathcal{T}\)?
- Write the strict upper level set of \(f_x\) at \(a\) as a section of a subset of \(X\times Y\).
- For the diagonal in \(\mathbb{R}^2\), what are \(\Delta_x\) and \(\Delta^y\)?
- Does the theorem state that every set with measurable sections is product-measurable? Explain the direction of the implication.