Editing System of imprimitivity (section)

== Induced representations ==

If ''X'' is a Borel ''G'' space and ''x'' ∈ ''X'', then the fixed point subgroup

:<math> G_x = \{g \in G: g \cdot x = x \} </math>

is a closed subgroup of ''G''. Since we are only assuming the action of ''G'' on ''X'' is Borel, this fact is non-trivial. To prove it, one can use the fact that a standard Borel ''G''-space can be imbedded into a compact ''G''-space in which the action is continuous.

''Theorem''. Suppose ''G'' acts on ''X'' transitively. Then there is a σ-finite quasi-invariant measure μ on ''X'' which is unique up to measure equivalence (that is any two such measures have the same sets of measure zero).

If Φ is a strict unitary cocycle

:<math> \Phi: G \times X \rightarrow \operatorname{U}(H) </math>

then the restriction of Φ to the fixed point subgroup ''G''<sub>''x''</sub> is a Borel measurable unitary representation ''U'' of ''G''<sub>''x''</sub> on ''H'' (Here U(''H'') has the [[strong operator topology]]). However, it is known that a Borel measurable unitary representation is equal almost everywhere (with respect to Haar measure) to a strongly continuous unitary representation. This restriction mapping sets up a fundamental correspondence:

''Theorem''. Suppose ''G'' acts on ''X'' transitively with quasi-invariant measure μ. There is a bijection from unitary equivalence classes of systems of imprimitivity of (''G'', ''X'') and unitary equivalence classes of representation of ''G''<sub>''x''</sub>.

Moreover, this bijection preserves irreducibility, that is a system of imprimitivity of (''G'', ''X'') is irreducible if and only if the corresponding representation of ''G''<sub>''x''</sub> is irreducible.

Given a representation ''V'' of ''G''<sub>''x''</sub> the corresponding representation of ''G'' is called the ''representation induced by'' ''V''.

See theorem 6.2 of (Varadarajan, 1985).