University of Birmingham > Talks@bham > Analysis Seminar > The Steinhaus-Weil property: its converse, Solecki amenability and subcontinuity

The Steinhaus-Weil property: its converse, Solecki amenability and subcontinuity

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The Steinhaus-Weil theorem that concerns us is the `interior points’ property—that in a topological group a non-negligible set S has the identity as an interior point of S⁻¹S. There are various converses; the one that mainly concerns us is due to Simmons and Mospan. Here the group is locally compact, so we have a Haar reference measure η. The Simmons-Mospan theorem states that a (regular Borel) measure has such a Steinhaus-Weil property if and only if it is absolutely continuous with respect to the Haar measure. We exploit the connection between the interior points property and a selective form of infinitesimal invariance afforded by a certain family of selective reference measures σ, drawing on Solecki’s amenability at 1 (and using Fuller’s notion of subcontinuity. We may thereby develop a number of relatives of the Simmons-Mospan theorem. This has links with topologies of Weil type.

This talk is part of the Analysis Seminar series.

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