Measurability of functions in product spaces

Gowrisankaran, Kohur

doi:10.1090/s0002-9939-1972-0291403-x

Cited by 17 publications

(13 citation statements)

References 6 publications

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“…For the second, we have that (i) for fixed x, the mapping t\->T(t, x) is continuous, and (ii) for fixed t, the map xh-»T(ί, x) is Borel. The result follows from a theorem of [3].…”

Section: Proposition For Xe^ [X] Is Either Finite or Uncountable Compact And Perfectmentioning

confidence: 85%

Time evolution for infinitely many hard spheres

Alexander

1976

Commun.Math. Phys.

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We construct the time evolution for infinitely many particles in R v interacting by the hard-sphere potential \ 0 Because there are abundant examples of hard-sphere configurations with more than one solution to the Newtonian equations of motion, we introduce the concept of a regular solution, in which the growth of velocities and crowding of particles at infinity are limited. We prove that (1) regular solutions exist with probability one in every equilibrium state, and (2) any configuration of the infinite system is the initial point of at most one regular solution. Equilibrium states are invariant under the time-evolution.

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“…For the second, we have that (i) for fixed x, the mapping t\->T(t, x) is continuous, and (ii) for fixed t, the map xh-»T(ί, x) is Borel. The result follows from a theorem of [3].…”

Section: Proposition For Xe^ [X] Is Either Finite or Uncountable Compact And Perfectmentioning

confidence: 85%

Time evolution for infinitely many hard spheres

Alexander

1976

Commun.Math. Phys.

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“…(iii) Suppose Y is a separable metrizable space with a dense sequence {y n } t n = 1, 2, For positive integers n and k let B nk be the open ball centre y n of radius 1/fc If fe ^b(X x V) we may represent [8]…”

Section: •Jmentioning

confidence: 99%

Functions and measures on products spaces

Babiker¹,

Knowles

1985

Mathematika

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“…By Corollary 4, x ϵ ( t ) is also right continuous with respect to t for every ω ∈ Ω . Therefore, from Gowrisankaran (1972: Theorem 3) it follows that ( ω , t ) ↦ x ϵ ( t ) is measurable with respect to the product σ -algebra F ⊗ B ( false[ τ , T false] ) . On the other hand, u ϵ ( t ) is piecewise continuous in t and is constant with respect to ( Y 1 ( ω ) , … , Y T ( ω ) ) . Therefore, …”

Section: A1 Nominal Trajectory Under Specific Observationsmentioning

confidence: 83%

SACBP: Belief space planning for continuous-time dynamical systems via stochastic sequential action control

Nishimura

Schwager

2021

The International Journal of Robotics Research

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We propose a novel belief space planning technique for continuous dynamics by viewing the belief system as a hybrid dynamical system with time-driven switching. Our approach is based on the perturbation theory of differential equations and extends sequential action control to stochastic dynamics. The resulting algorithm, which we name SACBP, does not require discretization of spaces or time and synthesizes control signals in near real-time. SACBP is an anytime algorithm that can handle general parametric Bayesian filters under certain assumptions. We demonstrate the effectiveness of our approach in an active sensing scenario and a model-based Bayesian reinforcement learning problem. In these challenging problems, we show that the algorithm significantly outperforms other existing solution techniques including approximate dynamic programming and local trajectory optimization.

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Measurability of functions in product spaces

Abstract: Let/be a function on a product space XxY with values in a separable metrizable space such that it is measurable in one variable and continuous in the other. The joint measurability of such a function is proved under certain conditions on A" and T.

Cited by 17 publications

References 6 publications

Time evolution for infinitely many hard spheres

Time evolution for infinitely many hard spheres

Functions and measures on products spaces

SACBP: Belief space planning for continuous-time dynamical systems via stochastic sequential action control

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