 Solution

Choose any strictly positive pricing kernel $Z\in \mathcal{Z}$. The assumed componentwise expectations and the linearity of expectation giveThe 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. Both conclusions follow: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 $\mathbb{E}Z=1$ is assumed in this abstract market; that normalization follows only if a unit-priced unit-payoff cash asset is included.