Proposition 20

Tag 09L2, permanent. Cite it as [Bourbaki, Tag 09L2].

— We conserve the hypotheses and notations of Prop. 19. Let $p \in [1, +\infty[$ and $g \in L^p(G, \beta)$ . Then

g = \lim_V g *^\beta f_V = \lim_V f_V *^\beta g

in the sense of the norm $N_p$ , the limit being taken with respect to the section filter of $\mathscr{B}$ .

It suffices to apply Prop. 6 (iii), and §2, No. 7, Cor. 3 of Lemma 4.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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