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Introduction

From [1], $$H(x) = \int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta\{dt\}.$$

From [2], $$\int_{X} f(y) \, \mathrm{d} \delta_x (y) = f(x).$$

Questions

Question 1. Does the $\mathbf{R}$ indicate integration along the entire real line? Or can it be any interval within the reals, or union of intervals?

Question 2. With reference to the top-most equation, in English, how should I understand $\delta\{dt\}$?

Question 3. With reference to the bottom-most equation, in English, how should I understand $\mathrm{d} \delta_x (y)$?

Question 4. How should I understand the differences betweeen $\delta\{dt\}$ and $\mathrm{d} \delta_x (y)$?

Bibliography

[1] https://en.wikipedia.org/wiki/Dirac_delta_function

[2] https://en.wikipedia.org/wiki/Dirac_measure

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You've run into the fact that the literature hasn't completely agreed on the notation for integration with respect to a measure.

$\int_{\mathbb{R}} f \delta\{dt\}$ is, indeed, an integral over the entire real line with respect to the dirac measure. If the author wanted to integrate elsewhere, they'd need to write something else.

As for your second, third and fourth questions, $\delta\{dt\}$ and $d\delta_x(y)$ are two different notations for the measure we integrate with respect to. The latter notation indicates in which point we have our mass, whereas simply $\delta$ omits this because we implicitly assume the mass to be at $x=0$ (I'm not trying to claim that this is particularly good notation). In the case of a general measure space, $X$, we don't typically have a favourite point, so we need to write where the mass is every time.

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