Helly's selection theorem

Abstract: We analyze the strength of Helly's selection theorem HST, which is the most important compactness theorem on the space of functions of bounded variation. For this we utilize a new representation of this space intermediate between $L_1$ and the Sobolev space W1,1, compatible with the, so called, weak* topology. We obtain that HST is instance-wise equivalent to the Bolzano-Weierstra\ss\ principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice.

Abstract: We present a constructive probabilistic proof of the fact that if B=(Bt)t≥0 is standard Brownian motion started at 0, and μ is a given probability measure on R such that μ({0})=0, then there exists a unique left-continuous increasing function b:(0,∞)→R∪{+∞} and a unique left-continuous decreasing function c:(0,∞)→R∪{−∞} such that B stopped at τb,c=inf{t>0|Bt≥b(t) or Bt≤c(t)} has the law μ. The method of proof relies upon weak convergence arguments arising from Helly’s selection theorem and makes use of the Levy metric which appears to be novel in the context of embedding theorems. We show that τb,c is minimal in the sense of Monroe so that the stopped process Bτb,c=(Bt∧τb,c)t≥0 satisfies natural uniform integrability conditions expressed in terms of μ. We also show that τb,c has the smallest truncated expectation among all stopping times that embed μ into B. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.