# Can the integration of integrable sections of a measurable function of two variables ever result in a non-measurable function?

I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct a counterexample refuting a result that had already been published in a peer-reviewed article.

Problem. Find measure spaces $(X,\mathcal{S},\mu)$ and $(Y,\mathcal{T},\nu)$, at least one of which is not $\sigma$-finite, and an $(\mathcal{S} \otimes \mathcal{T})$-measurable function $f: X \times Y \to \mathbb{R}_{\geq 0}$ with the following properties:

• The function $f(x,\bullet): Y \to \mathbb{R}_{\geq 0}$ belongs to ${L^{1}}(Y,\mathcal{T},\nu)$ for every $x \in X$.
• The function $\left\{ \begin{matrix} X & \to & \mathbb{R}_{\geq 0} \\ x & \mapsto & \displaystyle \int_{Y} f(x,\bullet) ~ \mathrm{d}{\nu} \end{matrix} \right\}$ is not $\mathcal{S}$-measurable.

Does anyone know if this problem can even be solved? Thank you very much for your time!

• Doesn't seem like the measure $\mu$ is even used here. – Daniel McLaury Oct 30 '16 at 21:56
• A short note to anyone studying this problem: In accordance with Daniel’s comment, one may let $\mu$ be the trivial measure on $(X,\mathcal{S})$, which makes $(X,\mathcal{S},\mu)$ a $\sigma$-finite measure space. Therefore, in order for a counterexample to exist, it is necessary that $(Y,\mathcal{T},\nu)$ be non-$\sigma$-finite, otherwise the second function in my post would be $\mathcal{S}$-measurable (as can be seen by analyzing any standard proof of the Fubini-Tonelli Theorem). – Transcendental Oct 31 '16 at 16:49
• Another short note: PhoemueX has provided a nice counterexample below that satisfies the second property, but it is unclear whether or not it also satisfies the first property. A brief analysis of PhoemueX’s argument leads one to the following question: Does there exist a Borel subset $M$ of $\mathbb{R}^{2}$ such that (i) each vertical cross-section of $M$ is finite, i.e., $\{ y \in \mathbb{R} \mid (x,y) \in M \}$ is finite for each $x \in \mathbb{R}$, and (ii) the projection of $M$ onto the first coordinate is a non-Borel analytic subset of $\mathbb{R}$? – Transcendental Nov 1 '16 at 7:01

Let $\mathcal{B}$ denote the class of Borel subsets of $[0,1]$ and let $A \subseteq [0, 1]$ be a non Borel set. Let $f$ be the characteristic function of the graph of a bijection from $A$ to $[0, 1]$. Then $f$ is $\mathcal{B} \otimes \mathcal{P}([0, 1])$-measurable (check) and the map $x \mapsto \int f(x, y) d\mu(y)$ is non-zero precisely on $A$ where $\mu$ is counting measure.

Edit: I was asked to provide more details so here they are.

Suppose $W \subseteq \mathbb{R}^2$ is such that every horizontal section $W^y = \{x : (x, y) \in W\}$ is closed. Then $W \in \mathcal{B} \otimes \mathcal{P}(\mathbb{R})$. To see this, define, for each interval $J$ with rational endpoints, $Y_J = \{y : W^y \cap J = \phi\}$. As each $W^y$ is closed, we have $$W = \mathbb{R}^2 \Big\backslash \bigcup \{J \times Y_J : J \text{ is an interval with rational endpoints}\} \in \mathcal{B} \otimes \mathcal{P}(\mathbb{R}).$$ It follows that the graph of every partial bijection on $\mathbb{R}$ is in $\mathcal{B} \otimes \mathcal{P}(\mathbb{R})$.

Now choose any non Borel set $A \subseteq \mathbb{R}$ and an injection $i: A \to \mathbb{R}$. Define $f: \mathbb{R}^2 \to \mathbb{R}$ to be the characteristic function of the graph of $i$. Let $\mu$ be the counting measure on $\mathbb{R}$. The map $x \mapsto \int f(x, y) d\mu(y)$ is precisely the characteristic function of $A$ and hence is non-Borel.

• Hi rumpf. Thank you for your response. Unfortunately, I don’t consider myself to be smart enough to check your claim in the third line. Could you elaborate further? – Transcendental Nov 1 '16 at 23:04
• For each rational interval $J$, let $Y_J$ be the set of points where the horizontal section of the graph of the bijection is disjoint with $J$. Now consider the union of $J \times Y_J$'s. – drumpf Nov 1 '16 at 23:12
• I am getting confused by these definitions; I think it would help if more things had names. Let's call $\phi : A \to [0,1]$ the desired bijection. It looks to me like you want $Y_J = \phi(A \cap J^c)$? That doesn't seem to make sense. In particular if $y = \phi(x)$ then $(x,y)$ is not in $J \times Y_J$ for any $J$. – Nate Eldredge Nov 2 '16 at 5:11
• @Nate: drumpf’s improved argument does appear to check out. What do you think? – Transcendental Nov 2 '16 at 18:26
• @Nate: It also appears that his argument works if we let $X$ be any uncountable second-countable $T_{1}$ topological space whose Borel $\sigma$-algebra $\mathcal{S}$ isn’t all of $\mathcal{P}(X)$ (i.e., $X$ has a non-Borel subset). Any uncountable standard Borel space automatically satisfies this property (assuming the Axiom of Choice, of course). – Transcendental Nov 2 '16 at 22:02

EDIT: The following only provides a partial answer, since it is not clear at all that the first property of the question ($f(x, \cdot) \in L^1(Y, \mathcal{T}, \nu)$) is fulfilled for the given example.

Let $(X, \mathcal{S})$ and $(Y, \mathcal{T})$ both be the real line with the Borel sigma algebra. Note that the product sigma algebra is again the Borel sigma algebra (but on $\Bbb{R}^2$).

It is well-known (see Projection of Borel set from $R^2$ to $R^1$) that not every projection of a Borel set is a Borel set. Hence, let $M \subset \Bbb{R}^2$ be a Borel set such that the projection $\pi_1 (M)$ is not Borel measurable.

Let $\mu$ be the counting measure on the real line. If $$F(x) := \int_{\Bbb{R}} 1_M (x,y) d \mu(y)$$ was measurable, then so would be the set $$\pi_1 (M) = \{x \,:\, \exists y : (x,y) \in M\} = \{x \,:\, F(x) > 0\}.$$

• Wikipedia claims that Fubini always works for the "maximal product measure." I wonder what that means if the iterated integrals aren't even defined. en.wikipedia.org/wiki/… – Christian Remling Oct 31 '16 at 18:50
• @Christian: Fubini’s Theorem states that if $(X,\mathcal{S},\mu)$ and $(Y,\mathcal{T},\nu)$ are measure spaces, and if $f \in {L^{1}}(X \times Y,\mathcal{S} \otimes \mathcal{T},(\mu \otimes \nu)_{\text{max}})$, where $(\mu \otimes \nu)_{\text{max}}$ denotes the maximal product measure, then the iterated integrals exist and are equal to $\displaystyle \int_{X \times Y} f ~ \mathrm{d}{(\mu \otimes \nu)_{\text{max}}}$. I suspect that the function $\mathbf{1}_{M}$ defined by PhoemueX above isn’t integrable on $\mathbb{R}^{2}$ with respect to the maximal product measure. – Transcendental Oct 31 '16 at 21:38
• @PhoemueX: Your counterexample clearly satisfies the second property, but can you show that it also satisfies the first property? In other words, can you prove that $\{ y \in \mathbb{R} \mid (x,y) \in M \}$ is finite for each $x \in \mathbb{R}$? Thanks! – Transcendental Nov 1 '16 at 7:04
• @Transcendental: Yes, I realized this problem just this morning. With the current construction, I don't think this is true. I will delete my answer if I see no way to fix it until this evening :( – PhoemueX Nov 1 '16 at 7:32
• As discussed here, the Lusin-Novikov theorem says that your example cannot satisfy the first property. – Nate Eldredge Nov 2 '16 at 15:18