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Canonical static concept record
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A topological space equipped with the -algebra generated by its open sets.
The Borel -algebra is canonically determined by the topology.
wp-boreltao-measure-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Take the smallest -algebra containing every open set. This measurable structure is canonical for the topology.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-borelwp-top-spacetao-measure-theoryEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Lebesgue measure starts from the Borel measure agreeing with interval and box volume and is completed by including subsets of measure-zero sets.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
tao-measure-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Retain only the Borel -algebra as a measurable-space structure.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-borelwp-measurabletao-measure-theory