A representation of a groupoid of mappings

Endowing functions from a set $X$ with values in a groupoid $G$ with the pointwise groupoid operations one obtains a groupoid which is called current groupoid. In this paper, we show that every unitary representation $L$ of a groupoid $G$ on a measurable Hilbert bundle lift to a unitary representation $pi_L$ of the current groupoid $C(X,G)$ on a frame groupoid of a Hilbert field.