Corollary 1

Tag 08RL, permanent. Cite it as [Bourbaki, Tag 08RL].

Integration, IV, p. 29, § 3. LP spaces

Let $(f_n)$ be a decreasing sequence of functions $\geqslant 0$ in $\mathscr{L}^p$ ; then, the lower envelope $f$ of the sequence belongs to $\mathscr{L}^p$ , the sequence $(f_n)$ converges in mean of order $p$ to $f$ , and

N_p(f) = \lim_{n \to \infty} N_p(f_n) = \inf_n N_p(f_n).

The first two assertions follow from Th. 5 applied to the sequence $g_n = f_1 - f_n$ , which is increasing and bounded above; the rest is then obvious.

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

Cited by

Corollary 3, § 3

Cites

Th. 5