Separable measure space | Wikipedia audio article
Channel: wikipedia tts
Date: 03/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Sigma-algebra


00:01:18 1 Motivation
00:02:32 1.1 Measure
00:02:51 1.2 Limits of sets
00:03:58 1.3 Sub σ-algebras
00:06:30 2 Definition and properties
00:09:26 2.1 Definition
00:09:35 2.2 Dynkin's π-λ theorem
00:11:43 2.3 Combining σ-algebras
00:13:50 2.4 σ-algebras for subspaces
00:18:01 2.5 Relation to σ-ring
00:18:31 2.6 Typographic note
00:19:12 3 Particular cases and examples
00:19:45 3.1 Separable σ-algebras
00:19:55 3.2 Simple set-based examples
00:21:51 3.3 Stopping time sigma-algebras
00:22:42 4 σ-algebras generated by families of sets
00:23:51 4.1 σ-algebra generated by an arbitrary family
00:24:02 4.2 σ-algebra generated by a function
00:24:51 4.3 Borel and Lebesgue σ-algebras
00:25:38 4.4 Product σ-algebra
00:28:52 4.5 σ-algebra generated by cylinder sets
00:29:19 4.6 σ-algebra generated by random variable or vector
00:30:07 4.7 σ-algebra generated by a stochastic process
00:33:50 5 See also
00:39:39 6 References
00:40:52 7 External links
00:42:54 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.9675180267393007
Voice name: en-AU-Wavenet-C


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that
includes X itself, is closed under complement, and is closed under  countable unions.
The definition implies that it also includes 
the empty subset and that it is closed under countable intersections.
The pair (X, Σ) is called a measurable space or Borel space.
A σ-algebra is a type of algebra of sets. An algebra of sets needs only to be closed under the union or intersection of finitely many subsets, which is a weaker condition.The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.
In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic, particularly when the statistic is a function or a random process and the notion of conditional density is not applicable.
If X = {a, b, c, d}, one possible σ-algebra on X is Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }, where ∅ is the empty set. In general, a finite algebra is always a σ-algebra.
If {A1, A2, A3, …} is a countable partition of X then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.
A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved - the σ-algebra produced by this process is known as the Borel algebra on the real line, and can also be conceived as the smallest (i.e. "coarsest") σ-algebra containing all the open sets, or equivalently containing all the closed sets. It is foundational to measure theory, and therefore modern probability theory, and a related construction known as the Borel hierarchy is of relevance to descriptive set theory.