A note on spanning with options

This note investigates the spanning power of ordinary options in spaces of contingent claims defined over uncountable many states of nature. It is proved that options might complete the space of Borel measurable and p-integrable contingent claims as well as the space of Borel measurable payoffs defined over a compact space. This work also shows that options fail to complete the space of continuous claims defined over a completely regular noncompact state-space but might complete the space of Borel measurable and p-integrable contingent claims and the larger space of Borel measurable payoffs.