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In person
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- IMAG Functional Analysis Seminar
- Title: Two classical concepts under the Ky Fan metric
- By Willian Franca (Federal University of Juiz de Fora, Brazil)
- Abstract: In this talk, we study measurable functions modulo equality almost everywhere via the Ky Fan metric, a standard metric that encodes convergence in measure. Two themes guide the talk. First, we present the notion of boundedness in measure and explain how it isolates the largest linear subspace of $L_0$ that still carries a compatible topological vector space structure; we also discuss completeness and algebraic properties under pointwise multiplication. Second, on a measure space equipped with a measurable exhaustion, we consider convergence in measure at infinity. We present criteria guaranteeing that an ``asymptotic value'' (when it exists) is well-defined, and we outline some properties of the subspace of $L_0$ consisting of functions that admit a limit at infinity. In the atomic setting we describe how certain numerical invariants of the measure control when these spaces become locally convex, Fr\'echet, or even Banach, and we indicate how familiar classical function and sequence spaces fit into this metric framework.
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