0

Can one apply the rule of addition of ordinary integrands for distributions? I mean, is it correct that, $\int_{a}^{b}f(x)\delta(x)\,dx\,=\,\int_{a}^{c}f(x)\delta(x)\,dx+\int_{c}^{b}f(x)\delta(x)\,dx$ ? The crucial point is, of course, when $c=0$, one seems to get $f(0)=2f(0)$.

Is there any explanation for that?

Yair
  • 499
  • For distributions, the integrals that you wrote are not defined. What is defined is $\langle \delta, \varphi\rangle = \varphi(0)$ when $\varphi$ is a test function. – Gribouillis Feb 23 '23 at 12:30
  • Well.. in my case the test function is simply 1. – Yair Feb 23 '23 at 12:31
  • $1$ is not a test function because it doesn't have a compact support. Nor is $1_{(a, b)}$ which is not ${\cal C}^\infty$, nor is $f(x)1_{(a, b)}$ unless the support of $f$ is included in $(a, b)$. – Gribouillis Feb 23 '23 at 12:32
  • Where can I read what is a valid test function? Will this rule be satisfied then? – Yair Feb 23 '23 at 12:36
  • 1
    A valid test function is an indefinitely differentiable function with compact support. This is defined in any course about distribution theory. – Gribouillis Feb 23 '23 at 12:38
  • Ok, thanks. For such a proper test function, is the rule above of splitting integrals is valid? btw, in physics we often use delta functions in a more abusive way, without this test function. – Yair Feb 23 '23 at 12:53

1 Answers1

2

For a distribution $g$ in general, the expression $\int_a^b g(x)\,\varphi(x)\,\mathrm d x$ has no meaning in general. It could be interpreted as $\langle g, \varphi\,\mathbf 1_{[a,b]}\rangle$ but then the test functions $\varphi\,\mathbf 1_{[a,b]}$ are not continuous in general.

In the particular case of the Dirac delta however, it can be defined in several ways, i.e. as a measure (see e.g. here). Both coincide when the test functions are smooth with compact support. Any distribution $\mu$ of order $0$ is a measure, and for such a measure one can define the integral with respect to it which verifies $$ \langle\mu,\varphi\rangle = \int \varphi(x)\,\mu(\mathrm d x) $$ for any $\varphi$ that is measurable with respect to $\mu$. In the particular case of the Dirac delta, it correspond to ask that $\varphi$ has a well defined value at $0$. This is in contrast with the case of the Lebesgue measure where integrable functions are only defined almost everywhere. Hence, for the Dirac delta, $\mathbf 1_{[0,b]}$ is different from $\mathbf 1_{(0,b]}$, since one has value $1$ at $0$ and the other has value $0$. Hence $$ \int_{(0,b]} \varphi(x)\,\delta(\mathrm d x) = \langle \delta,\varphi\,\mathbf 1_{(0,b]}\rangle = \varphi(0)\,\mathbf 1_{(0,b]}(0) = 0 $$ while $$ \int_{[0,b]} \varphi(x)\,\delta(\mathrm d x) = \langle \delta,\varphi\,\mathbf 1_{[0,b]}\rangle = \varphi(0)\,\mathbf 1_{[0,b]}(0) = \varphi(0). $$

More generally, for distributions of order $1$ (that is measures) one can split integrals in the following ways $$ \begin{align*} \int_{[a,b]} \mu(\mathrm d x) &= \int_{[a,c)} \mu(\mathrm d x) + \int_{[c,b]} \mu(\mathrm d x) \\ &= \int_{[a,c]} \mu(\mathrm d x) + \int_{(c,b]} \mu(\mathrm d x) \\ &= \int_{[a,c)} \mu(\mathrm d x) + \int_{(c,b]} \mu(\mathrm d x) + \mu(\{c\}) \end{align*} $$ where $\mu(\{c\})$ is the measure of the set $\{c\}$ by $\mu$. In the case of the Lebesgue measure, a singleton is always of measure $0$ so $\int_{[a,b]}$ and $\int_{(a,b)}$ give the same result, denoted as $\int_a^b$. In the case of the Dirac delta, $\delta(\{0\}) = 1$.

LL 3.14
  • 13,938
  • Thank you LL 3.14 for a clear explanation! – Yair Feb 23 '23 at 13:15
  • What about Fubini's theorem for exchanging the order of integrations, is it valid for distributions? – Yair Feb 23 '23 at 13:25
  • 1
    Yes, see e.g. https://mathoverflow.net/questions/72450/can-distribution-theory-be-developed-riemann-free/351028 (or the reference book on Distributions: "Théorie des Distributions" by L. Schwartz). In this case, the Fubini theorem holds for the general brackets for all the distributions. In the case of measures, this is a well-known result see e.g. https://math.stackexchange.com/questions/4092245/proper-usage-of-the-theorems-of-fubini-and-tonelli – LL 3.14 Feb 23 '23 at 22:04
  • Thanks you again! – Yair Feb 24 '23 at 00:22