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  • Measure space
    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
  • Sigma-algebra
    * {{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
  • Measure Theory
    ==Measures== ==Extending to measures==
    4 KB (733 words) - 01:41, 28 March 2015
  • Additive function
    ...s to include everything the old page did, link to propositions and lead to measures}}
    6 KB (971 words) - 18:16, 20 March 2016
  • Pre-measure
    {{Extra Maths}} A ''pre-measure''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Measure|measure
    5 KB (782 words) - 01:49, 26 July 2015
  • Measure
    * [[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
  • Measurable space
    A ''measurable space''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Tuple|tuple]] c
    2 KB (248 words) - 13:05, 2 February 2017
  • Lebesgue measure
    ...}(\mathbb{R}^n))\rightarrow\overline{\mathbb{R}_{\ge 0} }</math><ref>P27 - Measures, Integrals and Martingales - Rene L. Schilling</ref> that assigns every hal
    804 B (129 words) - 00:28, 20 December 2016
  • Measurable map
    ...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
  • Probability space
    ...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
  • Probability measure
    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
  • Semi-ring of half-closed-half-open intervals
    ...[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
  • Site projects:Patrolling measure theory/Task list
    * [[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
  • Pre-image sigma-algebra
    {{Todo|Measures Integrals and Martingales - page 16}}
    1 KB (177 words) - 22:12, 19 April 2016
  • Dynkin system generated by
    ...'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
  • Pre-measure/Properties in common with measure
    ...ref group="Note">Sometimes stated as ''monotone'' (it is ''monotone'' in ''Measures, Integrals and Martingales'' in fact!)</ref> if {{M|A\subseteq B}} then {{M
    1 KB (201 words) - 22:30, 30 March 2016
  • A function is a measure iff it measures the empty set as 0, disjoint sets add, and it is continuous from below (with equiv. conditions)
    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
  • Conditions for a Dynkin system to be a sigma-algebra
    ...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 Stochastics
    782 B (123 words) - 22:56, 2 August 2015
  • Conditions for a generated Dynkin system to be a sigma-algebra
    ...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
  • A collection of subsets is a sigma-algebra iff it is a Dynkin system and closed under finite intersections
    ...is a [[sigma-algebra|{{Sigma|algebra}}]] ''if and only if''<ref name="MIM">Measures, Integrals and Martingales</ref><ref name="PAS">Probability and Stochastics
    1 KB (239 words) - 13:21, 17 December 2016
  • Borel sigma-algebra generated by
    ...ets]] of a [[Topological space|topological space]], that is<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref>: (where {{M|(X,\mathca
    2 KB (244 words) - 08:30, 6 August 2015
  • Borel sigma-algebra
    ...f> [[Sigma-algebra|{{sigma|algebra}}]] on {{M|\mathbb{R} }}<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref>. It is [[Sigma-algebra
    5 KB (854 words) - 09:25, 6 August 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
    ==[[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
  • Limit of increasing sequence of sets
    ...n:Limit_of_Increasing_Sequence_of_Sets - taken from book I left at home! - Measures, integrals and Martingales</ref><ref name="PAS">Probability and Stochastics * {{M|A_n\uparrow A}}<ref>Measures, Integrals and Martingales - CHECK THIS REF</ref>
    2 KB (334 words) - 17:52, 21 August 2015
  • Pre-measure/New page
    ...similar thing to a [[measure]], and then shown what other authors call pre-measures extend uniquely to my kind of pre-measure. They also call my kind of pre-me ** [[Extending pre-measures to outer-measures]] as a part of
    3 KB (422 words) - 21:25, 17 August 2016
  • Index of notation/P
    | Sometimes used for [[Probability measure|probability measures]], the notation {{M|\mathbb{P} }} is recommended for these.
    1 KB (184 words) - 23:58, 11 January 2016
  • Notes:Measures
    # [[Books:Measures, Integrals and Martingales - René L. Schilling]] ===Measures, Integrals and Martingales===
    5 KB (771 words) - 03:06, 21 March 2016
  • Notes:Halmos measure theory skeleton
    ...st the core that let us get from [[pre-measure|pre-measures]] to [[measure|measures]] we would be left with what I call the "skeleton".
    4 KB (581 words) - 20:31, 3 April 2016
  • Notes:Measure theory plan
    * '''OUTER-MEASURE''' ({{M|\mu^*}}) - a construct named because it measures from the outside of a set and comes down (the inf), this lets us "measure" ...r measures are only ''subadditive'' not ''additive'' so they're not really measures. Make sure this weakness is demonstrated.
    4 KB (619 words) - 19:28, 24 May 2016
  • Site projects:Measure theory plan
    ...to what Halmos does but introduce definitions found elsewhere, for example Measures, Integrals and Martingales introduces a ''pre-measure'' on an algebra of se ...bras and sigma-algebras at the forefront of their minds when dealing with *measures
    2 KB (309 words) - 15:31, 27 March 2016
  • Measure theory terminology doctrine
    ...B}} there are 3 steps to constructing a [[measure]] (as in the thing that "measures" on a [[sigma-ring|{{sigma|ring}}]])
    2 KB (232 words) - 03:57, 30 March 2016
  • Extending pre-measures to outer-measures
    #*#* 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
  • Notes:Outer-measures to measures
    ...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
  • Notes:The set of all mu*-measurable sets is a ring
    ...y the [[subadditive]] property of {{M|\mu*}} (of all [[outer-measure|outer-measures]] we see: I've also consulted Measures Integrals and Martingales, it has the same thing in it. Both books say (and
    4 KB (828 words) - 03:11, 30 May 2016
  • Notes:Measures - Real and Abstract Analysis - Hewitt & Stromberg
    A continuation of the [[Notes:Measures]] series, covering the terminology used in [[Books:Real and Abstract Analys
    452 B (71 words) - 18:04, 31 May 2016
  • Notes:Generalising the limit
    ...bstraction as "converges in a group" makes little sense, and using it with measures would ''at most'' involve turning the measure into a metric, and I already
    6 KB (1,118 words) - 11:34, 30 July 2016
  • Extending pre-measures to measures
    # A [[extending pre-measures to outer-measures|pre-measure, {{M|\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge
    2 KB (257 words) - 17:27, 17 August 2016
  • Semi-ring of sets
    ...ng of sets]] instead). Then we can apply [[extending pre-measures to outer-measures]] to obtain an [[outer-measure]], all without going through the tedious tas
    3 KB (508 words) - 17:25, 18 August 2016
  • Types of set algebras/Type table
    * Is a [[Dynkin system]]<ref name="MIM">Measures Integrals and Martingales - Rene L. Schilling</ref><ref group="Note">Trivia
    4 KB (573 words) - 20:00, 19 August 2016
  • Doctrine:Measure theory terminology
    ===Measures=== \text{(other partial measures)} \ar@/_5ex/[urr] & & & \text{Outer measure} \ar@/_3ex/[ur] \\
    2 KB (333 words) - 17:49, 27 February 2017
  • Doctrine:Measure theory terminology/Proposals
    ...r in a way which preserves the measure of {{M|Y}}. That is, the sum of the measures of the spliced parts is the measure of {{M|Y}}. ...outer one (unless the context implies otherwise, for example if only inner-measures are in play) [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:29, 20 August
    3 KB (466 words) - 21:29, 20 August 2016
  • Outer splicing set
    ...ng}}]]. This observation is a critical part of [[extending pre-measures to measures]].
    2 KB (378 words) - 22:09, 20 August 2016
  • Geometric distribution
    ...eo}(1-p)}} in our terminology, they would write {{M|\text{Geo}(p)}}, which measures "trials until first failure" instead of success as we do
    3 KB (557 words) - 15:14, 16 January 2018
  • Alec's ordered data test
    ...ig)_{i\eq 1}^n}} which are of the ''ordered type'' in [[Alec's taxonomy of measures]], meaning we have the {{M|<}} and {{M|>}} operators forming a [[total ord
    1 KB (215 words) - 16:53, 14 November 2017
  • Alec's taxonomy of units
    Should the practicality of measures bleed in here?
    4 KB (592 words) - 07:29, 11 December 2017
  • Poisson race distribution
    * {{Caveat|Remember that Poisson measures stuff ''per unit thing''}}, meaning, for example, that say we are talking "
    6 KB (1,141 words) - 10:33, 24 December 2018
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