# Relationship between differential entropy and quantized entropy

I am reading Elements of Information Theory (2nd Ed.) by Cover. I have something that I couldn't figure out.

On page 248, theorem 8.3.1

If the density f(x) of the random variable X is Riemann integrable, then $$H(X^Δ)+log(⁡Δ)→h(f)=h(X)$$, as $$Δ→0$$.
Thus, the entropy of an n-bit quantization of a continuous random variable X is approximately h(X) + n.

The continuous random variable X is divided into bins of length $$\Delta$$.
$$X^Δ=x_i\quad if\quad i\Delta\le X<(i+1)\Delta$$. Where $$\Delta f(x_i)=\int_{i\Delta}^{(i+1)\Delta}f(x)dx$$.
The book gives a proof of the theorem, but what I didn't understand is how the last conclusion is reached. Clearly, if the range of X is [0, 1), the length of the bin $$\Delta$$ is related to n. $$-log_2(\Delta)=n$$ if the range is [0, 1). But this condition is not required in the theorem. Of course, if the range is finite, when $$\Delta$$ approaches 0, $$-log_2(\Delta) \rightarrow n$$. Also, the equation only holds when $$\Delta$$ is close to 0, which is also not stated in the conclusion. ("Thus, the entropy of an n-bit quantization of a continuous random variable X is approximately h(X) + n.")

Did I miss something, or is there some other proofs of that statement?

Thanks a lot.

In the book, by a "$$n-$$bit quantization" they mean how many binary (fractional) digits of precision we have.
When we speak in decimal base, we say we have "5 decimal digits" of precision (or that 5 decimal fractional digits are meaningful) when we quantize the real line in intervals of length $$1/10000=1/10^5$$. (Granted, the word precision is also used with different meanings; often it means the amount of significant digits). Similarly, to have 5 bits of precision (or a $$5-$$bit quantization) we must quantize the real range in intervals of length $$\Delta=1/2^5$$. Hence, indeed, $$-\log_2 \Delta = n$$.