Dirac Delta distribution of Theta function

Question

Edit: $\delta$ is the Dirac distribution and $\Theta$ is the unit step function.

In my theoretical physics studies I came across the following identity, that is supposed to be valid for all $t \in \mathbb{R}$

\begin{equation} \int \limits_{-\infty}^t \exp(t'-t) \delta(t - t') \mathrm{d}t'= \frac{1}{2} \end{equation}

which I could (contextually) make sense of as something I had been taught in the first semester

\begin{equation} \int \limits_{-\infty}^{\infty} \Theta(t'-t) \delta(t - t') \mathrm{d}t'= \frac{1}{2} \end{equation}

which is seriously riddled with inconsistency issues.

This sounds just like normal theoretical physics reasoning to me, but is there a way to formalize it?
Can we see the Dirac distribution as an element of a dual space of discontinuous functions in a proper sense? - I suspect not.
Or is it perhaps possible to introduce a kind of measure space where the Dirac measure can be seen to average at jump discontinuities? - I believe this should be possible in one dimension, but I am not sure.

Answer 1

A very simple way to think about it, which is not too far from how the Dirac delta is formally defined, is that any integral in dx of a product in which the delta(x-c) is one of the terms, is the value of the remaining terms of the product at c. So, in your case, your equation becomes $e(t)=\frac{1}{2}$ or $t=-\ln(2)$.

Answer 2

We know that $$\int_{-\infty}^\infty \exp(t'-t)\delta(t-t')\,dt'=\exp(0)=1$$ for any $t$, since $\int_{-\infty}^\infty f(x)\delta(x)\,dx=\delta(f)=f(0)$ for $f\in\mathscr{D}(\Bbb{R})$ and, even though $\exp\notin \mathscr{D}(\Bbb{R})$, we can pretend it is. What is more, $\delta(t-t')=0$ for $t'\ne t$ (or, to use the more official lingo, the support of $\delta$ is $\{0\}$), so really the integrand in the integral above is concentrated on $\{t\}$. It seems plausible that half the integral's value is in the half-integral below $t$, $\int_{-\infty}^t$, and the other half is in the half-integral above $t$, $\int_t^{\infty}$. That gives us the desired ${1\over2}$.

I don't quite know how to justify the integral with $\Theta$ in it.

Comment A. The official story on $\delta$ is conveniently found in Rudin Functional Analysis ch. 6.

Comment B. A closed question recently asked "Is there another Mathematics?" For me, the calcuations in Dirac's Principles of Quantum Mechanics and books of that ilk constitute another mathematics, which can be compared to the stuff in Rudin and in Reed and Simon, which is the true mathematics.