Proposition 4

— Let X and Y be two locally compact subspaces of T such that Y ⊂ X. If θ is a complex measure on T, then the measure (θ_X)_Y induced by θ_X on Y is equal to θ_Y ('transitivity of induced measures').

It suffices to observe that if g is an element of $\mathscr{H}(Y; \mathbf{C})$, then the extension by 0 of g to T may be obtained by extending by 0 the extension by 0 of g to X, or again, making use of the identifications of the Scholium, that $\varphi_Y · \theta = \varphi_Y(\varphi_X · \theta)$ (§5, No. 4, Prop. 8).