TITLE: Non-zero Conditional Differential Entropy between a random variable and a function of it
QUESTION [16 upvotes]: Let two continuous random variables, where the one is a function of the other: $X\,  $ and $\,  Y=g\left(X\right)$. Their mutual information is defined as
$$I\left(X,Y\right)\,=\,h\left(X\right)\,-\,h\left(X|Y\right)=\,h\left(Y\right)\,-\,h\left(Y|X\right)$$
where lowercase h denotes differential entropy, the entropy concept for continuous rv's. It is a proven fact that the mutual information between two variables, for discrete as well as for continuous rv's, is non-negative, and becomes zero only when the two rv's are independent (clearly not our case).
Using the fact that Y is a function of X we have
$$I\left(X,g\left(X\right)\right)\,=\,h\left(g\left(X\right)\right)\,-\,h\left(g\left(X\right)|X\right) \gt\,0\,\Rightarrow \,h\left(g\left(X\right)\right)\,\gt \,h\left(g\left(X\right)|X\right)$$
Now, differential entropy (unlike entropy for discrete rv's) can take negative values.
Assume that it so happens that $h\left(g\left(X\right)\right)\lt\,0$. Then from the positivity of mutual information we obtain
$$0\,\gt \,h\left(g\left(X\right)\right)\,\gt \,h\left(g\left(X\right)|X\right) \Rightarrow\; h\left(g\left(X\right)|X\right)\neq\,0$$
And this is the counter-intuitive puzzle: for any discrete random variable Z we always have $h\left(g\left(Z\right)|Z\right)\,=\,0$. This is intuitive: if Z is known, then any function of Z is completely determined -no entropy, no uncertainty remains, and so the conditional entropy measure is zero. But we just saw that, when dealing with continuous rv's where the one is a function of the other, their conditional differential entropy may be non-zero (it doesn't matter whether it is positive or negative), which is not intuitive at all. Because, even in this strange world of continuous rv's, knowing X, completely determines Y=g(X). I have searched high and low to find any discussion, comment or exposition of the matter, but I found nothing. Cover & Thomas book does not mention it, other books do not mention it, a myriad of scientific papers or web sites do not mention it.
My motives: a) Scientific curiosity. b) I want to use the concept of mutual information for continuous rv's in an econometrics paper I am writing, and I feel very uncomfortable to just mention the "non-zero conditional differential entropy" case without being able to discuss it a bit. So any intuition, reference, suggestion, idea, or full answer of course, would be greatly appreciated. Thanks.

REPLY [4 votes]: Differential entropy is only defined for absolutely continuous random variables with respect to Lebesgue measure. The joint distribution of
$$(X,g(X))$$
is not absolutely continuous because its support is
$$\{(x,g(x)):x\in\mathbb{R}\},$$
a measure 0 subset of $\mathbb{R}^2$.
The definition of conditional entropy requires an absolutely continuous joint distribution, $f(x,y)$: 
\begin{align} h(X|Y) &= -\int f(x,y) \log f(x|y)\ dxdy \\
                &= h(X,Y)-h(Y).
\end{align}
If $(X,Y)$ is not absolutely continuous against Lebesgue measure in $\mathbb{R}^2$, then there is no function $f(x,y)$ such that the probability measure of $(X,Y)$ can be expressed as an integral against $f$.
Therefore neither the quantity $h(g(X)|X)$, nor the quantity $I(X;g(X))$ are well defined as differential entropies.