I am reading this PDF talking about microcanonical ensemble : https://itp.uni-frankfurt.de/~valenti/WS13-14/all_1314_chap8.pdf

In this ensemble I know that the energy is fixed at a given $E$, we don't allow the energy to vary within an ensemble.

But, when he computes averages in this ensemble, the author allows the energy to be within $[E;E+\Delta]$. Is this $\Delta$ here for technical reasons ?

It is for example at the beginning of the paragraph 8.2 :

$$ <O>=\frac{1}{\Gamma(E,V,N)} \int \int_{E<H(p,q)<E+\Delta} d^{3N}q d^{3N}p ~ O(p,q) $$

With :

$$\Gamma(E,V,N)= \Delta \int \int d^{3N}q d^{3N}p \delta(E-H(p,q))$$

From my perspective, I would rather define :

$$<O>=\frac{1}{\Omega(E,V,N)} \int \int d^{3N}q d^{3N}p \delta(E-H(p,q)) O(p,q)$$

Where :

$$\Omega(E,V,N)=\int \int d^{3N}q d^{3N}p \delta(E-H(p,q)) $$

So for me I agree with their definition if we take $\Delta$ going to $0$, but for any non $0$ $\Delta$, the result is $\Delta$ dependant.

So why is there this $\Delta$ ? Is it only for technical purpose ? But why don't we define things like I proposed ? Am I wrong somewhere ?