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Let $\delta$ be rigorously defined as a generalized function (lim of a function).

I am guessing that $\int_{-1}^0\delta(x)d x=\int_0^1\delta(x)d x=\frac{1}{2}$?

Also, let $E$ denote a set contains 1/3 of the randomly chosen points on $[-1,1]$

Is it true that $\int_E\delta(x)d x=\frac{1}3$?

I must be wrong somewhere. For example it is possible that in a uncountable set I cannot just randomly choose points.

High GPA
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    $\int_{-1}^0\delta(x)d x$ would also need to be rigorously defined here: in particular is the integration over the interval $[-1,0]$ or $[-1,0)$ or $(-1,0]$ or $(-1,0)$ – Henry May 12 '22 at 08:44
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    Yes, an answer with this point of view can be found here https://math.stackexchange.com/questions/4427947/what-is-the-value-of-the-integral-int-inftya-deltax-a-dx-and-relate/4427952#4427952 – LL 3.14 Jan 26 '23 at 21:26

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As discussed several times in this community (for example, see this), the value of $\int_{0}^{a} \delta(x) \, \mathrm{d}x$ is not well-defined. There are several different choices of giving value to this expression, depending on the ambient physical process.

Also, there is no well-defined notion of a "set containing $\frac{1}{3}$ of the randomly chosen points on $[-1, 1]$". One may try to realize such a set via approximation: Let $E_n$ be a random set that is the union of "about $\frac{1}{3}$ of the subintervals of $[-1, 1]$ of equal length $2/n$. For example,

  • We may let each subinterval to be included in $E_n$ with probability $\frac{1}{3}$, or
  • We may choose $E_n$ to be one of $\binom{n}{\lfloor n/3\rfloor}$ possible unions of $\lfloor n/3\rfloor$ subintervals with equal probability.

Now, for any such sensible choices of approximations $E_n$, the corresponding indicator function $\mathbf{1}_{E_n}$ converges weakly to $\frac{1}{3}\mathbf{1}_{[-1, 1]}$. Of course, this is not an indicator function of a subset of $\mathbb{R}$.

Sangchul Lee
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  • Many thanks for your link and explanation. I in fact did a thorough search on this site: for example https://math.stackexchange.com/questions/1965974/integration-of-dirac-delta-function-from-3-to-0; https://math.stackexchange.com/questions/2881662/dirac-delta-in-polar-coordinates-integrating-delta-from-0-to-infty; https://math.stackexchange.com/questions/3166992/integrate-convolution-of-a-dirac-delta-function-from-0-to-t. But I did not find your link. Your link is much more helpful than the QA that I found! – High GPA May 13 '22 at 02:16