3

The Dirac measure is defined by $$\delta_x(A)= \begin{cases} 1 &\text{if $x \in A$}\\ 0 &\text{if $x \notin A$}\\ \end{cases}$$

Now, since $$\int f(y) \, d\delta_x(y)=f(x)$$ where $f:X\rightarrow \mathbb{R}$ is a function. So if $f(x)=1, \forall x\in X$ then $$\int \, d\delta_x(y)=1$$ How does the result of the integral change if we consider the following change to the integral $$\int \, d(y\delta_x(y))$$

I am trying to understand the equation in this paper. $\alpha$ here is a matrix of values. I am confused because the plot $P$ in the paper looks like a curve but the value of the integral seems to equal to a one or some constant to me

enter image description here

This is the plot

enter image description here

gbd
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  • I think it is my problem but I'm not used to your notation. Could you please explicit what do you mean with $d(y\delta_x(y))$? – Gabrielek Mar 05 '21 at 14:12
  • @Gabrielek, I mean the same notation in the paper. Eq. 3 – gbd Mar 05 '21 at 14:13
  • How do you have a "plot that looks like a curve" for $P$ but $P$ takes a matrix as an input ? What are the axes ? – Flewer47 Mar 05 '21 at 14:17
  • @Flewer47, it is not a smooth curve. I just meant it is not constant. I have added the plot above. – gbd Mar 05 '21 at 14:20
  • Well, you say that $\alpha$ is a matrix, but in the plot it is clearly a 1-dimensional matrix, that is a (real) scalar.

    Regarding the integral, for a fixed $\alpha$, you integrate the energy values when the condition inside the Dirac measure is satisfied, this only means you only take values on the set where the $\alpha_{X,m}$ do have an associated energy equal to $E$ for a fixed $E\in[E_{min},E_{max}]$.

    For different values of $\alpha$, the values of $P$ can be different, there is nothing here that says otherwise.

     – Flewer47 Mar 05 '21 at 14:24
  • @Flewer47, Regarding $\alpha$, there is a set of energies for each $m$ and $x$. So alpha is actually a three dimensional matrix. But for a particular $m$ and $x$ it becomes a one dimensional matrix . I am not sure what you meaning by the rest of your comment. Could you elaborate please. – gbd Mar 05 '21 at 14:31
  • @Flewer47, How do you integrate the energy values? – gbd Mar 05 '21 at 14:36

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