Suppose that $ (V, F) $ It is a foliated variety. There are three equivalent approaches to the notion of *transversal measure* as described in this book (see pages 65-69). I would like to understand the last line of section 5.$ alpha $ where it is stated that one easily controls, as in the case of flows, that $ Lambda $ meets definition 2 & # 39; & # 39 ;. So the context is this: one starts with a closed current $ C $ (of degree $ = dim F $) positive in the direction of the sheet and with the help of this current defines a measure $ mu_U $ (locally in $ U $, the domain of the foliation table) in the set of plates by the fromula

$$ langle C, omega rangle = int Big ( int _ { pi} omega Big) d mu_U ( pi). $$

Once we have this measure we can define it. $ Lambda (B): = int Card (B cap pi) d mu_ {U} ( pi) $ for any transverse borel $ B $ (that is, subset of Borel $ B subset V $ such that for each sheet $ L $ of the foliation $ B cap L $ it is at most countable.

Why $ Lambda $ satisfy $ Lambda (B) = Lambda ( psi (B)) $ for any injection of Borel $ psi $ that keeps the leaves.

My guess is that somehow you must follow the condition of $ C $ be closed but I'm not sure how to perform the calculations (for example, one problem I found is that $ Lambda $ It is defined locally and I do not see how to go from $ U $ to another foliation letter $ U & # 39; $ (What can happen by general $ psi $).