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This question on math (dot) stackexchange (dot) com has inspired me to write my own (possibly somewhat tendentious?) version of the question.

What does the question mean? That was a topic of a lot of disagreement in the comments, mostly between me and David C. Ullrich. I'll present my views here.

Suppose $\mu$ is a (nonnegative) measure on Borel subsets of $\mathbb R$.

For present purposes, I will take "Lebesgue measure" to mean the restriction of Lebesgue measure to Borel sets.

Let $F(x) = \mu( (-\infty,x] )$.

Then the total variation of the measure $\mu$ coincides with $\lim\limits_{x\to\infty} F(x)$.

If $\mu$ happens to be absolutely continuous with respect to Lebesgue measure $m$ then the Radon–Nikodym derivative $d\mu/dm$ exists and is equal to $f = dF/dx$.

In general, the total variation of $f$ differs from that of $\mu$. For example, suppose $\mu$ is the familiar measure $$ f(x)\,dx = \frac 1 {\sqrt{2\pi}} e^{-x^2/2}\,dx. $$ Then the total variation of the measure $\mu$ is $1$ and the total variation of the function $f$ is $\sqrt{2/\pi\,} \ne 1.$

But now suppose $\mu$ is the measure that assigns $1$ to the set $\{0\}$ and $0$ to every Borel set disjoint from that.

In that case we get $F(x) = \begin{cases} 0 & \text{if } x<0, \\ 1 & \text{if } x\ge 0. \end{cases}$

As before, the total variation of $\mu$ is $\lim\limits_{x\to\infty} F(x) = 1.$.

This measure is not absolutely continuous with respect to $m$ and has no Radon–Nikodym derivative with respect to $m$.

But $F$ has a derivative of a different sort, which is Dirac's $\delta$. Thus $\delta$ is formally in a role like that of $f$ above. That function $f$ has its own total variation generally differing from that of $\mu$, so the "total variation" of $\delta$, if such a thing can make sense, would probably not be $1$. (But David C. Ullrich said that $\delta$ simply is a measure, so its total variation is $1$.)

The value of $\delta$ at $0$ is sometimes said to be "infinity", but this is a sort of "infinity" that admits multiplication by reals numbers, yielding a different "infinity". However, people do not generally attempt to treat of the value of this function, but only work with the function and values of integrals involving it. It seems to me it could make sense to say the total variation is $2$ times that particular "infinity", and that of (for example) $3\delta$ would be $6$ times that particular "infinity".

If one is to make sense of such things, any "total variation" of $\delta'$ would be a far larger "infinity".

Arithmetic of various kinds of infinite "numbers" has been made logically rigorous, including Cantor's infinite cardinals and infinite ordinals, and Robinson's nonstandard real numbers, and rational functions as an ordered field, etc. Can any sort of rigorous arithmetic of these total variations similarly be developed? (It need not literally speak of infinite "numbers".)

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    $\begingroup$ I'm not sure that trying to address this in terms of "large infinite numbers" is the right way to go. The solution to making sense of the Dirac delta and other distributions wasn't to invent a large number to play the role of $\delta(0)$; rather, it was to describe them using an entirely different kind of mathematical object (functionals). So if there is a sensible way to define the total variation of a distribution, I expect it won't take values in a set of "numbers", but a set with a very different sort of structure. $\endgroup$
    – Nate Eldredge
    Commented Aug 18, 2016 at 0:26
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    $\begingroup$ Much of the confusion stems from the fact that the total variation of a function is not the same as the total variation of a measure -- they have different definitions. But if(!) the function is of bounded variation, then its distributional derivative is a measure, and the total variation of the function is given by the total variation of this measure. Unfortunate, but the terminology is by now too deeply entrenched. (I believe there's a special case for functions of bounded variation from $\mathbb{R}$ to $\mathbb{R}$, where both definitions coincide for the function(!).) $\endgroup$
    – Christian Clason
    Commented Aug 18, 2016 at 11:24
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    $\begingroup$ Also, $\delta'$ is not a (Radon) measure, since its pairing with a continuous function is not well-defined -- it's a proper distribution (but can be identified with an element of the dual space of $C^1$). Hence its total variation (in either sense) is not defined. You could, of course, define something like the "higher order total variation" (in the sense of functions) of $F$ via duality (following the definition of total variation for functions -- this is actually done), but you'd have to prove yourself that it satisfies the desired (e.g., norm) properties. $\endgroup$
    – Christian Clason
    Commented Aug 18, 2016 at 11:32
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    $\begingroup$ Cortizo says "It is shown that theories already presented as rigorous mathematical formalizations of widespread manipulations of Dirac’s delta function are all unsatisfactory, and a new alternative is proposed. I think by "unsatisfactory" he means something like "incomplete" in the sense that they fall short of justifying all of the things that physicists routinely do with these "functions". $\qquad$ $\endgroup$
    – Michael Hardy
    Commented Aug 19, 2016 at 0:50
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    $\begingroup$ Interesting article, and of course he uses the Poisson kernel (see the Wiki article), related to electrostatics, as a nascent Dirac delta. // Zemanian doesn't use the term nascent, but presents similar functions. Scan the introductory chapter. (It was years ago that I looked at the book, so I can't give the pages.) $\endgroup$
    – Tom Copeland
    Commented Aug 19, 2016 at 1:05

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