Metric isomorphism

of two measure spaces and ; also "strict isomorphism" or "point isomorphism"

2020 Mathematics Subject Classification: Primary: 28Axx [MSN][ZBL]

A bijective mapping for which images and inverse images of measurable sets are measurable and have the same measure (here is some Boolean -algebra or -ring of subsets of the set , called measurable, and is a given measure on ). There is the more general notion of a (metric) homomorphism of these spaces (called also measure-preserving transformation), that is, a mapping such that inverse images of measurable sets are measurable and have the same measure. For , 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 , , , and let be a metric isomorphism; then it is said that 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 (subsets, functions, transformations, and systems of these) one can give a meaning to the assertion that under a metric isomorphism these objects transform into each other. It is then said that 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 of measure zero the corresponding objects may be considered as objects in (for transformations this means that the are invariant relative to these transformations, whereas for subsets and functions this makes sense for any : take the intersection of the considered subsets with or the restrictions of the functions to ) and that is a metric isomorphism of the objects . 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 are the Hilbert spaces 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 functions is not always in ), and the Boolean measure σ-algebras , obtained from 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 modulo 0 induces an isomorphism of the Boolean measure algebras and a unitary isomorphism of the Hilbert spaces which is also multiplicative, that is, takes a product (whenever defined) to the product of the images of the multiplicands. If is a Lebesgue space, then the converse is true: Every isomorphism of the Boolean measure -algebras , or every multiplicative unitary isomorphism of the spaces , is induced by some metric isomorphism modulo 0.

References

[R]	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

[H]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802