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Conditional expectation as an ae class
Definition
Assume AC for the supplied existence theorem. Write for the unique class in consisting of conditional-expectation versions of . A chosen real -measurable representative is a version. Equalities and inequalities involving these classes mean almost-sure equalities and inequalities.
Existence is Conditional expectation exists by radon nikodym and uniqueness is Conditional expectation is unique almost surely, applied to Conditional expectation given a sigma algebra. The quotient is The space as the quotient by null functions. If almost surely, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree gives identical event integrals, so the output class is independent of the input representative. The The Axiom of Choice assumption is inherited from RN existence.
Source notes
Durrett §4.1, printed p.206, version/uniqueness convention; van der Vaart Definition 1.1 and Theorem 1.3, printed pp.1–2.
Depends on
- Conditional expectation given a sigma algebra
- Conditional expectation exists by radon nikodym
- Conditional expectation is unique almost surely
- The space $L^p(\mu)$ as the quotient by null functions
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- The Axiom of Choice
Used by
- Conditional cauchy schwarz inequality Corollary
- A submartingale need not have increasing sample paths Counterexample
- A version can fail a pointwise identity on a null set Counterexample
- An unbounded predictable transform may lose integrability Counterexample
- Conditioning does not preserve strict inequalities Counterexample
- Taking out an unbounded factor needs integrability Counterexample
- Conditional expectation for nonnegative variables Definition
- Conditional probability given a sigma algebra Definition
- Conditional variance Definition
- Martingale difference sequence Definition
- Martingale submartingale and supermartingale Definition
- Predictable quadratic variation in discrete time Definition
- Conditional expectation given a discrete random variable Example
- Conditioning on a finite partition Example
- Dyadic conditional expectation martingale Example
- L² best prediction by conditional expectation Example
- Likelihood ratio martingale Example
- Polya urn proportion martingale Example
- Conditional expectation process is a martingale Lemma
- Conditioning a known variable and an independent variable Lemma
- Conditional expectation is a class not a canonical pointwise function Remark
- Basic algebra and order properties of conditional expectation Theorem
- Conditional fatou and dominated convergence Theorem
- Conditional jensen inequality Theorem
- Doob decomposition of an integrable adapted process Theorem
- Taking out what is known Theorem
- Tower property of conditional expectation Theorem
- Uniform integrability of conditional expectations of one variable Theorem
Dependency tree · two levels
20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Durrett, Probability: Theory and Examples, 5th ed. (standard reference, not scraped)