  We define a classical probability analogue of Voiculescu's free entropy
dimension that we shall call the classical probability entropy dimension of a
probability measure on $\mathbb{R}^n$. We show that the classical probability
entropy dimension of a measure is related with diverse other notions of
dimension. First, it can be viewed as a kind of fractal dimension. Second, if
one extends Bochner's inequalities to a measure by requiring that microstates
around this measure asymptotically satisfy the classical Bochner's
inequalities, then we show that the classical probability entropy dimension
controls the rate of increase of optimal constants in Bochner's inequality for
a measure regularized by convolution with the Gaussian law as the
regularization is removed. We introduce a free analogue of the Bochner
inequality and study the related free entropy dimension quantity. We show that
it is greater or equal to the non-microstates free entropy dimension.
