Characterizations of (sequentially) order continuous linear forms on vector lattices are given in terms of their behaviour relative to families of orthogonal elements. As a consequence, the non existence of real measurable cardinals can be characterized by the property that the sequentially order continuous and the order continuous linear forms on order complete vector lattices coincide. This gives rise to a counter example to a conjecture of [1].

In this article we give a simple proof of the existence of the quadratic variation process of a continuous local martingale by providing an explicit expression for it.