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Let $(M,g)$ be a closed, Riemannian manifold. Let $u_n$ be a sequence of smooth, positive functions on $M$ such that the $L^1(M)$ norm of the sequence is bounded uniformly. Can we say that $u_n$ converges tightly to a Radon measure on $M$?

I can't find a definition of tight convergence anywhere, but it is used in P. Lions "The Concentration Compactness Principle in the Calculus of Variations. The limit case, Part I". I'm guessing it means weak convergence with a tight limit, but I can't verify this with the web.

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Do you mean that a subsequence of $u_n$ converges tightly? – Mark Aug 10 '11 at 3:29
Matt: yes, a subsequence would be fine. – Q-the-curvature Aug 30 '11 at 1:18

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