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In [[mathematics]], in particular [[probability theory]], the '''Doob–Dynkin lemma''', named after [[Joseph L. Doob]] and [[Eugene Dynkin]], characterizes the situation when one [[random variable]] is a function of another by the [[subset|inclusion]] of the [[sigma algebra|<math>\sigma</math>-algebras]] generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being [[measurable function|measurable]] with respect to the <math>\sigma</math>-algebra generated by the other.
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The lemma plays an important role in the [[conditional expectation]] in probability theory, where it allows to replace the conditioning on a [[random variable]] by conditioning on the [[sigma-algebra|<math>\sigma</math>-algebra]] that is [[Sigma_algebra#.CF.83-algebra_generated_by_a_function|generated]] by the random variable.
 
==Statement of the lemma==
Let <math>\Omega</math> be a [[sample space]]. For a function <math>f:\Omega \rightarrow R^n</math>, the <math>\sigma</math>-algebra generated by <math>f</math> is defined as the family of sets <math>f^{-1}(S)</math>, where <math>S</math> are all [[Borel set]]s.
 
'''Lemma''' Let <math>X,Y: \Omega \rightarrow R^n</math> be [[random element]]s and <math>\sigma(X)</math> be the <math>\sigma</math> algebra generated by <math>X</math>. Then <math>Y</math> is <math>\sigma(X)</math>-[[measurable]] if and only if <math>Y=g(X)</math> for some [[Borel measurable]] function <math>g:R^n\rightarrow R^n</math>.
 
The "if" part of the lemma is simply the statement that the composition of two measurable functions is measurable. The "only if" part is the nontrivial one.
 
By definition, <math>Y</math> being <math>\sigma(X)</math>-[[measurable]] is the same as <math>Y^{-1}(S)\in \sigma(X)</math> for any Borel set <math>S</math>, which is the same as <math>\sigma(Y) \subset \sigma(X)</math>. So, the lemma can be rewritten in the following, equivalent form.
 
'''Lemma''' Let <math>X,Y: \Omega \rightarrow R^n</math> be random elements and <math>\sigma(X)</math> and <math>\sigma(Y)</math>  the <math>\sigma</math> algebras generated by <math>X</math> and <math>Y</math>, respectively. Then <math>Y=g(X)</math> for some Borel measurable function <math>g:R^n\rightarrow R^n</math> if and only if <math>\sigma(Y) \subset \sigma(X)</math>.
 
==References==
* A. Bobrowski: ''Functional analysis for probability and stochastic processes: an introduction'', Cambridge University Press (2005), ISBN 0-521-83166-0
* M. M. Rao, R. J. Swift : ''Probability Theory with Applications'', Mathematics and Its Applications, Band 582, Springer-Verlag (2006), ISBN 0-387-27730-7
 
{{DEFAULTSORT:Doob-Dynkin Lemma}}
[[Category:Probability theorems]]
[[Category:Measure theory]]

Latest revision as of 19:38, 3 August 2014



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It comes with a 4" x 24" belt with a speed of 1400 sfpm for rapidly material removal. Its housing has been developed to permit flush sanding, and changing the sanding belt is rapidly achieved by a release lever. An additional sander tool presented by Milwaukee is the 6010 which is a sheet sander. This sheet sander is precision engineered for low vibration, a feature you will appreciate although sanding. Alternatively, the belt and disc mount directly to the motor's driveshaft.

If you move the tool as well rapidly into the belt it can catch - possibly destroying the belt, possibly pulling the tool from your hands and flipping it around the area. I have by no means had a dilemma with a belt but have received reports of even superior belts having problems. NOTE The belt sanders I own (3) are developed and sold to hobbyists who make occasional use the machine. The belt flexes as it passes around the wheels.

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