On deterministic and random fixed points

Tan, Kok-Keong; Yuan, Xian-Zhi

doi:10.1090/S0002-9939-1993-1169051-2

On deterministic and random fixed points
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Proc. Amer. Math. Soc. 119 (1993), 849-856 Request permission

Abstract:

Based on an extension of Aumann’s measurable selection theorem due to Leese, it is shown that each fixed point theorem for $F(\omega , \cdot )$ produces a random fixed point theorem for $F$ provided the $\sigma$-algebra $\Sigma$ for $\Omega$ is a Suslin family and $F$ has a measurable graph (in particular, when $F$ is random continuous with closed values and $X$ is a separable metric space). As applications and illustrations, some random fixed points in the literature are obtained or extended.

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