Definition 0.1. A groupoid representation induced by measure can be defined as measure
induced operators or as operators induced by a measure preserving map in the context
of Haar systems with measure associated with locally compact groupoids, Glc. Thus, let
us consider a locally compact groupoid Glc endowed with an associated Haar system
ν = 
, and μ a quasi-invariant measure on UGlc. Moreover, let (X1,𝔅1,μ1) and
(X2,𝔅2,μ2) be measure spaces and denote by L0(X
1) and L0(X
2) the corresponding spaces
of measurable functions (with values in ℂ). Let us also recall that with a measure-preserving
transformation T : X1→X2 one can define an operator induced by a measure preserving map,
UT : L0(X
2)→L0(X
1) as follows.
Next, let us define ν = ∫
νudμ(u) and also define ν−1 as the mapping x
x−1. With f ∈ C
c(Glc),
one can now define the measure induced operator Indμ(f) as an operator being defined on L2(ν−1)
by the formula:
Remark:
One can readily verify that :
,
and also that Indμ is a proper representation of Cc(Glc), in the sense that the latter is usually
defined for groupoids.
