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318 B (41 words) - 07:26, 27 April 2015
- is a [[Measure|measure]] ''if and only if''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> ##* This is called ''Continuity of measures from below''<ref name="MIM"/>2 KB (376 words) - 12:12, 27 July 2015
- ==[[The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set/Statement|Statement]]== {{:The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set/Statement}}4 KB (688 words) - 21:03, 31 July 2016
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267 B (42 words) - 05:47, 12 March 2016
- # [[Books:Measures, Integrals and Martingales - René L. Schilling]] ===Measures, Integrals and Martingales===5 KB (771 words) - 03:06, 21 March 2016
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801 B (129 words) - 20:44, 31 July 2016
- #*#* Using [[the (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set]], which states, symbolically: ...{{MSeq|A_n|post=\in\alpha_A}} and {{M|\beta_A}} is the sum of all the pre-measures {{WRT}} {{M|\bar{\mu} }} of the sequences of sets in {{M|\alpha_A}})11 KB (1,921 words) - 16:59, 17 August 2016
- ...as I have taken MIAM's notation of {{M|\bar{\mu} }} for [[pre-measure|pre-measures]]. Halmos uses {{M|\bar{\mu} }}</ref> defined by {{M|\bar{\mu}^*:A\mapsto\m ...rightarrow\bar{\mathbb{R} }_{\ge 0} }} is {{sigma|finite}} then so are the measures {{M|\bar{\mu}^*}} on {{M|\sigma(\mathcal{R})}} and {{M|\bar{S} }}.6 KB (1,067 words) - 22:19, 23 May 2016
- A continuation of the [[Notes:Measures]] series, covering the terminology used in [[Books:Real and Abstract Analys452 B (71 words) - 18:04, 31 May 2016
- # A [[extending pre-measures to outer-measures|pre-measure, {{M|\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge2 KB (257 words) - 17:27, 17 August 2016
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38 B (5 words) - 06:43, 11 December 2017
Page text matches
- A ''measure space''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Tuple|tuple]]:1 KB (188 words) - 15:24, 21 July 2015
- * {{M|X\in\mathcal{A} }}<ref group="Note">''Measures, Integrals and Martingales'' puts this in the definition of {{sigma|algebra ...e 28 - Springer - Graduate Texts in Mathematics - 18</ref><ref name="MIAM">Measures, Integrals and Martingales - Rene L. Schilling</ref>8 KB (1,306 words) - 01:49, 19 March 2016
- ==Measures== ==Extending to measures==4 KB (733 words) - 01:41, 28 March 2015
- ...s to include everything the old page did, link to propositions and lead to measures}}6 KB (971 words) - 18:16, 20 March 2016
- {{Extra Maths}} A ''pre-measure''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Measure|measure5 KB (782 words) - 01:49, 26 July 2015
- * [[A function is a measure iff it measures the empty set as 0, disjoint sets add, and it is continuous from below (wit ===Trivial measures===6 KB (941 words) - 14:39, 16 August 2016
- A ''measurable space''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Tuple|tuple]] c2 KB (248 words) - 13:05, 2 February 2017
- ...}(\mathbb{R}^n))\rightarrow\overline{\mathbb{R}_{\ge 0} }</math><ref>P27 - Measures, Integrals and Martingales - Rene L. Schilling</ref> that assigns every hal804 B (129 words) - 00:28, 20 December 2016
- ...lled '''<math>\mathcal{A}/\mathcal{A}'</math>-measurable'''<ref name="MIM">Measures, Integrals and Martingales - Rene Schilling</ref>, or '''<math>\mathcal{A}-5 KB (792 words) - 02:31, 3 August 2015
- ...al{A},\mu)}} where <math>\mu</math> is a [[Probability measure]]<ref>p22 - Measures, Integrals and Martingales - Rene L. Schilling</ref>, we recall that <math>2 KB (338 words) - 22:55, 2 May 2015
- We often denote probability measures using {{M|\mathbb{P} }}, the space this measures is implicit from the context.393 B (50 words) - 22:52, 2 May 2015
- ...[pre-measure]] we can follow the usual path of [[extending pre-measures to measures]] ...age 18 in [[Books:Measures, Integrals and Martingales - René L. Schilling|Measures, Integrals and Martingales - René L. Schilling]] if stuck.}}4 KB (680 words) - 00:23, 20 August 2016
- * [[A function is a measure iff it measures the empty set as 0, disjoint sets add, and it is continuous from below (wit * [[Measures are monotonic and subtractive]]5 KB (645 words) - 11:40, 21 August 2016
- {{Todo|Measures Integrals and Martingales - page 16}}1 KB (177 words) - 22:12, 19 April 2016
- ...'the [[Dynkin system]] generated by {{M|\mathcal{G} }}'' as<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref>:2 KB (245 words) - 15:16, 16 December 2016
- ...ref group="Note">Sometimes stated as ''monotone'' (it is ''monotone'' in ''Measures, Integrals and Martingales'' in fact!)</ref> if {{M|A\subseteq B}} then {{M1 KB (201 words) - 22:30, 30 March 2016
- is a [[Measure|measure]] ''if and only if''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> ##* This is called ''Continuity of measures from below''<ref name="MIM"/>2 KB (376 words) - 12:12, 27 July 2015
- ...ap}}-closed (which is to say it is also a {{M|p}}-system)]]<ref name="MIM">Measures, Integrals and Martingales</ref><ref name="PAS">Probability and Stochastics782 B (123 words) - 22:56, 2 August 2015
- ...thcal{G}\subseteq\mathcal{P}(X)}} is {{M|\cap}}-closed then<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> {{M|1=\delta(\mathcal{660 B (97 words) - 14:47, 27 July 2015
- ...is a [[sigma-algebra|{{Sigma|algebra}}]] ''if and only if''<ref name="MIM">Measures, Integrals and Martingales</ref><ref name="PAS">Probability and Stochastics1 KB (239 words) - 13:21, 17 December 2016
