I have modeled the Dirac delta function using limiting value of the rectangular pulse and now proving its area property. In the proof, I need to interchange the order of integration and limit, which does not look possible. I looked out at the similar problems on math stackexchange but could not find it anywhere. The details about the problem are as follows:
To model the delta function, first I took the rectangular pulse function as:
\begin{equation} p_n(t) = \frac{1}{n} \ \textrm{for}\ |t| \le \frac{n}{2}\ \textrm{otherwise}\ 0 \end{equation}
Based on this rectangular pulse, we can define the delta function as follows:
\begin{equation} \delta(t) = \lim_{n \rightarrow 0}{p_n(t)} \end{equation}
So the above equation precisely captures the behavior of the delta function.
Now, I want to prove the area property of the delta function, which is:
\begin{equation} \int_{-\infty}^{+\infty}{\delta(t) dt} = 1 \end{equation}
By writing the above equation using the definition of the delta function, it results into:
\begin{equation} \int_{-\infty}^{+\infty}{\lim_{n \rightarrow 0}{p_n(t)}}dt = 1 \end{equation}
Next, if we look at the integrand of the above integral, it is divergent as the function $\lim_{n \rightarrow 0}{p_n(t)}$ has infinite value, so overall its not integrable. Now, I want to swap the limit and integral operator, which is only possible if the limit is convergent (where it is divergent in this case).
My question is that is there really a way I can swap the order of this limit and integral? If yes then how would I do it?
Note: I also checked out many questions/answers regarding this on math stackexchage, but in all cases they directly take limit outside the integral, i.e., $\lim_{n \rightarrow 0}{\int}_{-\infty}^{+\infty}{p_n(t)} dt$, and it would only be equal to $\int_{-\infty}^{+\infty}{\delta(t) dt}$, if the limits and integral operators can be swapped. Moreover, the integral that I am using is the gauge integral.
I would highly appreciate your earliest response on it.
Thanks in advance.