Corollary 13.5.1143C93EF28279

Proof. For $t=0$ and $t=1$ this is the hypothesis. Otherwise, multilinearity (proposition 13.2.6) expands the left side as $\sum _{k=0}^ n\binom nk(1-t)^ kt^{n-k}\langle T^ k\wedge S^{n-k}\rangle $. Bounding the terms with $1\leq k\leq n-1$ by corollary 13.5.10 and the terms with $k=0,n$ by the hypotheses, we obtain

by the binomial formula. The measure on the right is finite, because $f$ and $g$ are $\nu $-integrable by the hypotheses and each density $f^{k/n}g^{(n-k)/n}$ is at most $\frac knf+\frac{n-k}ng\leq f+g$: for $k=0$ and $k=n$ this is an equality, and for $1\leq k\leq n-1$ it is (13.5.9). $\square$