Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 1 / a

If such $\rho$ existed, then integrability and $H\cdot X\ge 0$ would giveBut $\rho >0$ almost surely and $H\cdot X$ is nonnegative and strictly positive with positive probability. Hence $\rho (H\cdot X)$ is nonnegative and nonzero with positive probability, so its expectation is strictly positive. This contradiction proves that no such positive state-price density exists.