Sums and Products of Measurable Functions

Why Pairing the Functions Helps

For two measurable real-valued functions \(f\) and \(g\), the pointwise expressions \(f+g\) and \(fg\) involve both functions at once. Trying to describe their level sets directly can lead to complicated unions of conditions on \(f\) and \(g\). A simpler route is to regard the two functions together as one map into the plane. The sum and product are then obtained by applying continuous functions to that map.

Throughout, let \((X,\mathcal{M},\mu)\) be any measure space; no finiteness, sigma-finiteness, or completeness assumption on \(\mu\) is needed. A function \(f:X\to\mathbb{R}\) is measurable if \(f^{-1}(U)\in\mathcal{M}\) for every open set \(U\subseteq\mathbb{R}\), equivalently for every Borel set \(U\in\mathcal{B}_{\mathbb{R}}\). In this tutorial, \(f\) and \(g\) are real-valued everywhere, so their sum and product are also finite real-valued functions everywhere. Measurability statements below hold on all of \(X\), not merely almost everywhere.

The Pair Map Is Measurable

The Borel sigma-algebra \(\mathcal{B}_{\mathbb{R}^2}\) is generated by open rectangles \(I\times J\), where \(I,J\subseteq\mathbb{R}\) are open intervals. In fact, rectangles with rational endpoints form a countable base: every open subset of the plane is a union of the rational-endpoint rectangles it contains, and that union is countable. Thus it is enough to check preimages of open rectangles.

Proof. Let \(I,J\subseteq\mathbb{R}\) be open intervals. By the definition of the pair map, \(F^{-1}(I\times J)=\{x:f(x)\in I\text{ and }g(x)\in J\}=f^{-1}(I)\cap g^{-1}(J)\). Measurability of \(f\) and \(g\) gives \(f^{-1}(I),g^{-1}(J)\in\mathcal{M}\), so their intersection belongs to \(\mathcal{M}\). Every open subset of \(\mathbb{R}^2\) is a countable union of rational-endpoint open rectangles. Its preimage under \(F\) is therefore a countable union of measurable preimages of rectangles, and belongs to \(\mathcal{M}\). Since the Borel sigma-algebra is generated by open sets, the preimage of every Borel set is measurable. This proves that \(F\) is measurable. \(\square\)

Continuous Operations Give Measurable Functions

We use the composition rule: if \(F:(X,\mathcal{M})\to(Y,\mathcal{N})\) is measurable and \(T:(Y,\mathcal{N})\to(Z,\mathcal{P})\) is measurable, then \(T\circ F\) is measurable. Indeed, for each \(C\in\mathcal{P}\), \((T\circ F)^{-1}(C)=F^{-1}(T^{-1}(C))\); first \(T^{-1}(C)\in\mathcal{N}\), then its preimage under \(F\) belongs to \(\mathcal{M}\). In particular, a continuous map between Euclidean spaces is Borel-measurable, because the preimage of every open set under a continuous map is open.

Proof. Let \(F(x)=(f(x),g(x))\), which is measurable by the proposition. Define \(S,P:\mathbb{R}^2\to\mathbb{R}\) by \(S(u,v)=u+v\) and \(P(u,v)=uv\). Both maps are continuous. For addition, the estimate \(|S(u,v)-S(u_0,v_0)|\le |u-u_0|+|v-v_0|\) shows continuity at every \((u_0,v_0)\). For multiplication, if \(|u-u_0|<1\), then \(|u|\le |u_0|+1\), and \(|uv-u_0v_0|\le |u|\,|v-v_0|+|v_0|\,|u-u_0|\). For any desired positive error, choosing both coordinate differences sufficiently small makes this bound smaller than that error; this proves continuity of \(P\). By the composition rule, \(S\circ F\) and \(P\circ F\) are measurable. Their values at \(x\) are \(f(x)+g(x)\) and \(f(x)g(x)\), respectively. Hence \(f+g\) and \(fg\) are measurable everywhere on \(X\). \(\square\)

A Common Level-Set Mistake

A tempting but incorrect shortcut is to try to express the set where a sum exceeds a threshold using only whether each summand exceeds that threshold. For example, one might claim \(\{x:f(x)+g(x)>1\}=\{x:f(x)>1\}\cup\{x:g(x)>1\}\). Take \(X\) to consist of one point, with its full sigma-algebra, and set \(f=g=3/4\) at that point. The left side contains the point because \(3/4+3/4=3/2>1\), while the right side is empty because neither \(3/4>1\). The proposed identity is false.

The pair-map proof avoids the need for such an identity. For any real number \(c\), the set \(H_c=\{(u,v)\in\mathbb{R}^2:u+v>c\}\) is open because addition is continuous. Therefore \(\{x:f(x)+g(x)>c\}=F^{-1}(H_c)\in\mathcal{M}\). Likewise, \(\{(u,v):uv>c\}\) is open because multiplication is continuous, so its preimage under \(F\) is measurable. These are examples of the general theorem: the preimage under \(F\) of any Borel set in the plane is measurable.

The same reasoning applies on every measure space and does not use the measure \(\mu\) beyond identifying the underlying measurable space. It also explains the scope of the result: here the functions take finite real values, and addition and multiplication are ordinary continuous operations on \(\mathbb{R}^2\). Arithmetic involving infinite values requires separate definitions and care, so it is not part of this theorem.