Metric isomorphism
of two measure spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m0636601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m0636602.png" />; also "strict isomorphism" or "point isomorphism"
This category corresponds roughly to MSC {{{id}}} {{{title}}}; see {{{id}}} at MathSciNet and {{{id}}} at zbMATH.
A bijective mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m0636603.png" /> for which images and inverse images of measurable sets are measurable and have the same measure (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m0636604.png" /> is some Boolean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m0636605.png" />-algebra or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m0636606.png" />-ring of subsets of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m0636607.png" />, called measurable, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m0636608.png" /> is a given measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m0636609.png" />). There is the more general notion of a (metric) homomorphism of these spaces (called also measure-preserving transformation), that is, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366010.png" /> such that inverse images of measurable sets are measurable and have the same measure. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366011.png" />, instead of an isomorphism or a homomorphism one speaks of a (metric) automorphism or an endomorphism.
In correspondence with the usual tendency in measure theory to ignore sets of measure zero, there is (and is primarily used) a "modulo 0" version of all these ideas. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366014.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366015.png" /> be a metric isomorphism; then it is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366016.png" /> is an isomorphism modulo 0, or almost isomorphism of the original measure spaces (the stipulation "modulo 0" is often omitted).
For a number of objects given in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366017.png" /> (subsets, functions, transformations, and systems of these) one can give a meaning to the assertion that under a metric isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366018.png" /> these objects transform into each other. It is then said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366019.png" /> is a metric isomorphism of the corresponding objects. It is also possible to speak of their being metrically isomorphic modulo 0. This means that for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366020.png" /> of measure zero the corresponding objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366021.png" /> may be considered as objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366023.png" /> (for transformations this means that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366024.png" /> are invariant relative to these transformations, whereas for subsets and functions this makes sense for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366025.png" />: take the intersection of the considered subsets with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366026.png" /> or the restrictions of the functions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366027.png" />) and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366028.png" /> is a metric isomorphism of the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366029.png" />. A class of all objects metrically isomorphic modulo 0 to each other is called a (metric) type; two objects of this class are said to have the same type.
Associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366030.png" /> are the Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366031.png" /> in which, in addition to the usual Hilbert space structure, there is also the operation of ordinary multiplication of functions (defined, it is true, not everywhere, since the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366032.png" /> functions is not always in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366033.png" />), and the Boolean measure σ-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366035.png" />, obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366036.png" /> by identifying sets with symmetric difference of measure zero (that is, factorizing with respect to the ideal of sets of measure zero). A metric isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366037.png" /> modulo 0 induces an isomorphism of the Boolean measure algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366038.png" /> and a unitary isomorphism of the Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366039.png" /> which is also multiplicative, that is, takes a product (whenever defined) to the product of the images of the multiplicands. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366040.png" /> is a Lebesgue space, then the converse is true: Every isomorphism of the Boolean measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366041.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366042.png" />, or every multiplicative unitary isomorphism of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366043.png" />, is induced by some metric isomorphism modulo 0.
References
| [1] | V.A. Rokhlin, "On the fundamental ideas of measure theory" Mat. Sb. , 25 : 1 (1949) pp. 107–150 (In Russian) |
Comments
See also Ergodic theory for additional references. As a rule, the adjective "metric" is not anymore used and one simply speaks of an isomorphism of measure spaces, a homomorphism of measure spaces, etc.
References
| [2] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Template:ZBL |
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