A tomographic quantum measure for a multimode system is introduced. Symplectic tomograms describing quantum states of the system with many degrees of freedom are shown to be equal to partial derivatives of the von Neumann probability distribution functions of homodyne random variables. The central limit theorem known in quantum probability theory is applied to describe properties of the symplectic quantum measures introduced. An example of the centre-of-mass homodyne quadrature is studied in the context of the central limit theorem.