First some definitions.

For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the subbase consisting of the sets $\{\mu\in P(X):\mu(U)>a\}$ where $U$ is an open set in $X$ and $a\in\mathbb R$.

A subset $A$ of an Abelian Polish group $X$ is called Haar-null if there exists a measure $\mu\in P(X)$ such that $\mu(A+x)=0$ for all $x\in X$. In this case the set of test measures $$T(A)=\{\mu\in P(X):\forall x\in X\;\mu(A+x)=0\}$$ is not empty.

Dodos proved that for an analytic subset $A$ of an Abelian Polish group $X$ the following trichotomy holds:

$\bullet$ $T(A)$ is empty;

$\bullet$ $T(A)$ is meager and dense in $P(X)$;

$\bullet$ $T(A)$ is comeager in $P(X)$.

A subset $A$ of a Polish group $X$ is called generically Haar-null if its set of test measures $T(A)$ is comeager in $P(X)$. So, a generic measure on $X$ witnesses that $A$ is Haar-null.

Dodos proved that each $\sigma$-compact Haar-null set in an Abelian Polish group $X$ is generically Haar-null and each analytic generically Haar-null set in $X$ is meager and Haar-null.

On the other hand, a Borel subset of a locally compact Abelian Polish group is Haar-null if and only if its Haar measure is zero. In particular, a Borel subset of the real line is Haar-null if and only if its Lebesgue measure is zero.

To my own surprise I cannot neither prove nor disprove the following conjectures.

Conjecture 1. The real line contains a Borel generically Haar-null set, which cannot be covered by countably many compact Haar-null sets.

Conjecture 2. The real line contains a meager Haar-null Borel subset which is not generically Haar-null.


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