Talk:Dirac measure

I suggest that the element x in the definition of the Dirac measure be replaced by a different element (say, a), so that it is easy to distinguish it from the set X. Currently, the use of both the uppercase and the lowercase symbols is confusing, especially when one is used in subscripts.

I'll wait 3-4 days to see if there are any comments before making this change.

Arcenciel 05:20, 2 November 2005 (UTC)

is the Dirac measure really an atomic measure?

Consider a delta sequence D_n(x)and consider an integrable function f(.) such that f(0+) and f(0-) exist. Then the integrals of D_n(x)f(x)dx converge to [f(0+)+f(0-)]/2, not to f(0). In fact the value of f(0) is of no influence on the limit of these integrals.

Leocat 30 September 2006

This is not a contradiction. The delta sequence $D_{n}$ converges weakly (or vaguely). This means that, for every bounded continuous function (or cont. function with compact support) f, the sequence of integrals converges to the integral w.r.t. the limit-measure:

$\int f(x)D_{n}(x)\,dx\to \int f(x)\,d\delta _{0}(x)=f(0)$

But this need not be true for non-continuous functions. (More exactly, it is true for all functions which are continuous everywhere except on a set which is a null set for the limit measure. This is not the case for the function you mentioned.)--Trigamma 12:16, 15 January 2007 (UTC)

Restrictions on sigma-algebra

The current article says that any sigma-algebra does. However, since the singleton {x} is to be assigned a measure, ít clearly must be included in the sigma-algebra, right?

--Kaba3 (talk) 18:50, 27 August 2010 (UTC)

Aah.. I see, there is an error in the article when it claims that singleton {x} is assigned measure 1. Indeed, that singleton does not even need to be measurable. I'll fix this.

--Kaba3 (talk) 18:55, 27 August 2010 (UTC)

Merge with Dirac delta function?

The page on the Dirac delta function already has a section on the delta function as a measure. Isn't this page duplicative? 128.135.100.105 (talk) 11:20, 21 November 2016 (UTC)

Holds as a theorem of Lebesgue integration?

The expression,

$\int _{X}f(y)\delta _{x}(y)\,\mathrm {d} y=f(x),$,

is often referred to correctly as an abuse of notation. Despite its widespread use, it is mathematically confusing and misleading. Specifically, if $\,\mathrm {d} y\,$ is to be read as Lebesgue measure then the value of this integral is zero for all $f$, because the support of delta is a set of Lebesgue measure zero. So the question remains, what theorem of Lebesgue integration theory does this statement refer to and how was it intended to be read symbolically? I claim there is no justifiable notation for the quoted notation to make sense, and hence the correct reason it is dubbed an abuse of notation, because it is either redundant or contradictory. M. A. Maroun 13:57, 4 August 2020 (UTC) — Preceding unsigned comment added by MMmpds (talk • contribs)