then | ψ ψ ( x ) | 2 is the probability density function for a measurement of the particle's position at a given time, defined as the Radon–Nikodym derivative with respect to the Lebesgue measure (e.g. on the set R of all real numbers). As probability is a dimensionless quantity, | ψ (x) | must have the inverse dimension of the variable of integration x. For example, the above amplitude has dimension [L ], where L represents length.