If the mass contained in a finite volume V of fluid at a given instant is M then the average mass density of the fluid is simply M/V. In order to define the instantaneous mass density at a point in the fluid a more sophisticated approach must be adopted in view of the fact that a fluid in reality is composed of discrete particles. Thus if M is interpreted as the actual mass in V at a given instant, it should be apparent that the limit of M/V as V approaches zero does not exist since M is not a continuous function of V. This difficult is usually avoided by visualizing that the real fluid can be replaced by a continuum which displays the same macroscopic behavior as the real fluid. This approach, however, does not lead to any definite definition of density but instead leads to further inquires in regard to what is meant by the phrase ``displays the same macroscopic behavior as the real fluid.''
The approach adopted here is as follows. We will regard M not as the actual mass contained in the volume V at the instant t but as the most probable mass which we should expect to find in the volume V at the instant t. The expectation of finding this mass is greater for greater V, and for all practical purposes is 100 per cent if V is greater than about 1 cubic micron, in the case of sea water at normal pressure and temperature. The most probable value depends upon the statistical assemblage of molecules which make up the fluid.
It is then asserted that there exists a density function , such that
for any arbitrary volume of fluid. The evaluation of this function is another matter; needless to say, this is no small task.
In any measurement of density it is the mean value for a finite volume which is determined. If the volume is of sufficiently small macroscopic ize, the ample value of M/V represent and estimate of the idealized density function . From the rigorous standpoint, however, it is impossible to completely define the function through measurements, just as it is impossible to measure the exact pressure of a fluid at a point. From a statistical point of view, however, the confidence limits of the sample estimates of these quantities can be reduced to a very small fraction of the magnitude of the quantity measured by selecting an optimum scale of measurement which is on the borderline between microscopic and macroscopic. There must exist an irreducible minimum of confidence limits at soem scale of measurement; it is this scale which would define the optimum scale for measurement. If the sample is smaller than this scale, the lesser number of discrete particles in the sample leads to large confidence limits on the sample estimate from the statistical standpoint; if the sample is greater than the optimum scale then the greater size leads to too great a sample for identifying the mean property with the property at a particular point within the sample.