Markov kernel: Difference between revisions

Revision as of 16:36, 17 February 2020

Revision as of 18:16, 23 March 2020

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=== Measurable functions. ===

Take <math>(X, \mathcal{A})</math> and <math>(BY, \mathcal{B})</math> arbitrary measurable spaces, and let <math>f:X \to Y</math> be a measurable function. Now define <math>\kappa(dy|x) = \delta_{f(x)}(dy)</math> i.e.

:<math> \kappa(B|x) = \mathbf{1}_B(f(x)) = \mathbf{1}_{f^{-1}(B)}(x) = \begin{cases}1 & \text{if } f(x) \in B\\ 0 & \text{otherwise}\end{cases}</math> for all <math>B \in \mathcal{B}</math>.

Note that the indicator function <math>\mathbf{1}_{f^{-1}(B)}</math> is <math>\mathcal{A}</math>-measurable for all <math>B \in \mathcal{B}</math> iff <math>f</math> is measurable.