= Solution
Choose any strictly positive <pricing kernel> $Z\in\mathcal Z$. The assumed componentwise <expectations> and the <linearity of expectation> give
$$
\mathbb E[Z(H\cdot P)]=H\cdot\mathbb E[ZP]=H\cdot p\leq0.
$$
The integrand is nonnegative, so its <expectation> is also nonnegative and hence zero. A nonnegative <random variable> with zero <expectation> vanishes <almost surely>. Since $Z>0$ <almost surely>, this forces $H\cdot P=0$ <almost surely>. \b[Both conclusions follow]:
$$
\boxed{H\cdot p=0,\qquad H\cdot P=0\quad\text{almost surely}.}
$$
The strict positivity of the <pricing kernel> is essential: a kernel allowed to vanish could miss a positive payoff on its zero set. No normalization $\mathbb EZ=1$ is assumed in this abstract market; that normalization follows only if a unit-priced unit-payoff cash asset is included.
Back to article page
