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It is shown that, if  G  is an uncountable Polish group and  A  ⊆  G  is a universally measurable set such that  A  −1  A  is meager, then the set  T   l  ( A ) = {μ ∈  P ( G ): μ( gA ) = 0 for all  g  ∈  G } is co‐meager. In particular, if  A  is analytic and not left Haar‐null, then 1 ∈ Int ( A  −1  AA  −1  A ).

The Steinhaus property and Haar‐null sets