For stochastic functions over the product of a general space and a time interval, the standard type non-anticipating integration scheme is applied with respect to the stochastic measures with independent values. In this framework we suggest a non-anticipating stochastic derivative as a stochastic analogue of the Radon-Nykodim derivative. This stochastic derivative is then connected to the problem of the integral representation of random variables in a standard L2-space. Several specifications of the differentiation formula are derived in the case of Lévy stochastic measures.
Key words: measurable and predictable modification, stochastic integral, stochastic non-anticipating derivative, Lévy stochastic measure, Malliavin derivative, Clark-Haussmann-Ocone formula.