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Let be measurable spaces. Let be a mapping. is called a measurable function if i.e.Β the pre-images of all sets in the output Ο-algebra are in the Ο-algebra of the input space.
In the context of our analysis class we can examine this definition from the POV of topological spaces
Let be a measurable space and be a topological space. We say is measurable if
A Summary of MATH 891
Borel function
Ο-algebra containing all Borel subsets
Limit of Measurable Functions is Measurable
Measurability Criterion for Topological Codomain
Supremum & Infimum Preserve Measurability
HΓΆlder's Inequality
Jensen's Inequality
Minkowski's Inequality
Law
Lebesgue-Stieltjes Integral
Convolution
Lebesgue Integral
Change of Variable Formula
Beppo Levi Theorem
Dominated Convergence Theorem
Fatou's Lemma
Monotone Convergence Theorem
Lebesgue Integral is a Measure
Essential bound
Integrable
Integrals of Functions that are Zero a.e.
Simple Function
Closure of Measurability
Composition of Measurable Functions
Criterion for Measurability
Function of real measurable functions is measurable
Measurability of Continuous Functions
Pushforward Measure
Fubini-Tonelli
Slices of product measurable function are in measurable in resultant Ο-algebras
Total Variation
Admissible Policy
Measurable Selection Conditions
Controlled Markov Chain
Blackwell's Irrelevant Information Theorem
Discounted Cost
Belief MDP
Kalman Filter
Static Quadratic Team
Witsenhausen's Intrinsic Model
Norm-like Function
Conditional Expectation
Random Variable
(Ξ) Set of Predictable Locally Integrable Processes
Semimartingale
Bichteler-Dellacherie Theorem
Independence of Bounded RV
Left Continuous Adapted = Predictable
Replicating Portfolio
Disintegration
Feller Property
Local Martingale
Doob's Upcrossing Inequality
Martingale Convergence Theorem
Stochastic Kernel
Random Time
Ionescu Tulcea Theorem