 Differential entropy under an increasing transformation

Let $X$ have a probability density function on an open interval, and let $g$ be continuously differentiable, strictly increasing there, with $g^{\prime }(x)>0$. The change of variables formula givesIf the differential entropy and the expected absolute logarithm of $g^{\prime }(X)$ are finite, taking the expectation of the negative logarithm proves the displayed formula, for a fixed base $b>1$. Local stretching reduces probability density; its contribution to differential entropy is the expected logarithmic stretch. The map is invertible, so this change does not mean that the transformation destroyed information about $X$.