Definition (Measurable)
Let be measurable spaces. Let be a mapping. is called a measurable function if i.e. the pre-images of all sets in the output σ-algebra are in the σ-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
Integral Operator
A Summary of MATH 891
Borel function
Measurability Criterion for Topological Codomain
Supremum & Infimum Preserve Measurability
Hölder's Inequality
Jensen's Inequality
Minkowski's Inequality
Law
Lebesgue-Stieltjes Integral
Beppo Levi Theorem
Change of Variable Formula
Convolution
Dominated Convergence Theorem
Fatou's Lemma
Lebesgue Integral
Monotone Convergence Theorem
Infinity Norm
Integrable
Integrals of Functions that are Zero a.e.
Closure of Measurability
Composition of Measurable Functions
Criterion for Measurability
Simple Function
Pushforward Measure
Fubini Theorem (For indicator functions)
Fubini-Tonelli
Stein variational gradient descent
Norm-like Function
Conditional Expectation
Random Variable
Total Variation metric
Bichteler-Dellacherie Theorem
Itô Stochastic Integral is a Local Martingale
Predicable Processes
Semimartingale
Stopping Time Integral
Portfolio
Ionescu Tulcea Theorem
Disintegration
Stochastic Kernel
Doob's Upcrossing Inequality
Local Martingale
Martingale Convergence Theorem
Random Time
Blackwell's Irrelevant Information Theorem
Controlled Markov Chain
Discounted Infinite Horizon Optimization
Measurable Selection Conditions
Policy
Belief MDP
Kalman Filter
Finite Memory Approximations
LQG Teams
Witsenhausen's Intrinsic Model