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.