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pivoxa15
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What are some examples of sigma algebra operations?
matt grime said:Open and closed is a property of topological spaces. Topologies are in some sense completely different from sigma algebras. So, no, there is no need for open or closedness to have anything to do with sigma algebras, or anything else like it (D-algebras, etc). Of course, it is unlikely one would have been defined without the other, but that is does not stop there being no technical relation between the two, as opposed to a conceptual one. And in any 'real life' case, you will always be trying do measure theory on a topological space anyway.
A sigma algebra is a collection of subsets of a given set that satisfies certain properties. It is often used in probability theory and measure theory to define a set of events or sets of measure.
The three basic operations of sigma algebra are union, intersection, and complement. These operations can be used to generate new sets from existing ones within the sigma algebra.
One example of a sigma algebra is the Borel sigma algebra on the real line. It contains all open intervals, closed intervals, half-open intervals, and countable unions and intersections of these intervals.
In probability theory, sigma algebra operations are used to define the set of events for which probabilities can be assigned. The sigma algebra must contain all possible outcomes of an experiment and be closed under the three basic operations.
A sigma algebra must be closed under complement in order to ensure that the set of events is consistent and that all possible outcomes are accounted for. This property also allows for the easy calculation of probabilities using the complement rule.