Proposition 1

— For every real number $p \geqslant 1$ and every pair of positive functions $f, g$ (finite or not) defined on X,

(Minkowski’s inequality).

The inequality (1) is obvious when one of the terms of the second member is equal to $+\infty$. In the contrary case, $f$ and $g$ are finite almost everywhere ($§ 2$, No. 3, Prop. 7). If $f_1$ and $g_1$ are finite positive functions equivalent to $f$ and $g$, respectively, then $f_1^p, g_1^p$ and $(f_1 + g_1)^p$ are equivalent to $f^p, g^p$ and $(f + g)^p$, respectively, and since equivalent positive functions have the same upper integral ($§ 2$, No. 3, Prop. 6), we are reduced to proving the inequality (1) in the case that $f$ and $g$ are everywhere finite; but in this case the inequality is a special case of the general Minkowski inequality proved in Ch. I, No. 2, Prop. 3.

We shall also have occasion to make use of the following elementary inequality: if $p \geqslant 1$ then, for any numbers $a \geqslant 0,\ b \geqslant 0$,

The inequality is obvious if $a = b = 0$ or if one of the numbers $a, b$ is $+\infty$; if $a, b$ are finite and $a + b > 0$, it may be written

which follows from the fact that