Proposition 1

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

Integration, chapter IX, § 1. Premeasures and measures on a topological space

Let $(p_\alpha)_{\alpha \in A}$ be a family of encumbrances on T. The sum and upper envelope of the family $(p_\alpha)$ (in $\mathscr{F}_+(\mathscr{F}_+(T))$ ) are then encumbrances.

The sum of a finite family of encumbrances obviously being an encumbrance, it suffices to treat the case of the upper envelope. The properties a), b), c) of Definition 1 being obviously satisfied, it remains to establish d). Set $p = \sup_\alpha p_\alpha$ ; then, with the notations of Definition 1 d),

p(f) = \sup_\alpha p_\alpha(f) = \sup_\alpha \sup_n p_\alpha(f_n) = \sup_n \sup_\alpha p_\alpha(f_n) = \sup_n p(f_n).

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

Cited by

Definition 1, § 1
Definition 4, § 1

Cites

Definition 1