Corollary 5

— Let $f_1, f_2, \ldots, f_n$ be n positive integrable functions, and let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be n numbers $> 0$ such that $\sum_{i=1}^n \alpha_i = 1$; under these conditions, the function $f_1^{\alpha_1} f_2^{\alpha_2} \cdots f_n^{\alpha_n}$ is integrable and

For, the product $f_1^{\alpha_1} f_2^{\alpha_2} \cdots f_n^{\alpha_n}$ is measurable, being the product of measurable functions (§ 5, No. 3, Th. 1 and its Cor. 5); since the inequality (8) is true for upper integrals (Ch. I, No. 2, Cor. of Prop. 2), the function $f_1^{\alpha_1} f_2^{\alpha_2} \cdots f_n^{\alpha_n}$ is integrable (§ 5, No. 6, Th. 5), whence the corollary.

Cor. 2 of Th. 2 is sharpened by the following proposition: