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Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$.

Suppose we have a measurable function $f: G \rightarrow \mathbb{T}\cong \mathbb{R}/\mathbb{Z}$ such that $f(x + y) = f(x)+f(y)$ for all $(x, y)\in Z$, where $Z\subset G\times G$ with $(\mu\times\mu)(Z)=1$.

Question:

Can we find a continuous, group homomorphism $\phi: G \rightarrow \mathbb{T}\cong \mathbb{R}/\mathbb{Z}$, i.e., $\phi$ lies in the dual group of $G$, such that $\phi(x)=f(x)$ almost everywhere?


Remarks:

1, This is essentially a question asked here by someone else with some change. Since no answer appeared, I think it is OK to ask it again here. Note that $Z$ is not necessarily of product type.

2, It seems to be able to extend $f$ to a continuous, almost everywhere group homomorphism, but I do not see how to get a group homomorphism.(this claim seems Not true)

3, Any help, suggestions, references are appreciated, thanks in advance!

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2 Answers 2

In order to produce a continuous homomorphism, a good method is to take somehow the average of f. For example, you can define $$F(x)= \frac{1}{\mu(G)} \int _G(f(x+y)+f(-y))d\mu(y)$$ By assumption, for a.e. $x,y\in G$ we have $f(x+y)+f(-y)=f(x)$. Try to show that $F(x)=f(x)$ for a.e. $x\in G$ and $F$ is continuous.

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$Z\subset G\times G$ is not necessarily of product type. –  ougao Sep 6 at 1:49
    
The measure of $G\times G \setminus Z$ is zero if and only if the measure of the slices of $G\times G\setminus Z$ is zero, by Fubini. –  Dimitrios Ntalampekos Sep 6 at 4:25
    
"...iff the measure of the slices of $G\times G\Z$ is zero a.e., by Fubini". I still do not see how to use the assumption that $f(x+y)=f(x)+f(y), \forall (x,y)\in Z" to get $f(x+y)+f(-y)=f(x)$ for a.e. x, a.e. y in G. –  ougao Sep 6 at 12:33
1  
Isn't $\mu$ a probability measure? i.e. $\mu(G)=1$? –  Dimitrios Ntalampekos Sep 6 at 16:04
    
both are true. Is it related to my above comment? –  ougao Sep 6 at 19:39

With someone else's help, I learnt that this is essentially proved in the following paper

A.Kleppner, Measurable homomorphisms of locally compact groups, Proc. Amer. Math. Soc. 106(1989), no. 2, 391-395.

But I have not checked the proof.

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