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In this question we see that any linear operator in the vector space of smooth functions over $\mathbb{R}$ that satisfies the product rule and chain rule is the derivative operator. The set of all continuous functions $f$ is a vector space, and a linear operator on it is taking an antiderivative and then adding a constant to make $f(0) = 0.$ Is there any simple set of properties that uniquely determines this linear operator other than the standard definition?

mathlander
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  • So, basically, you ask what is the dual of C(R), correct? – Salcio Nov 30 '22 at 00:48
  • I'm asking how we can identify the function of integration from the dual of C(R). – mathlander Nov 30 '22 at 00:49
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    Since the dual of continuous compactly supported functions on R is the finite Borel measures, this is equivalent to asking for a set of properties which uniquely characterizes the Lebesgue measure. It turns out that any translation invariant Borel measure which satisfies $\mu([0,1]) = 1$ is in fact the Lebesgue measure. – kieransquared Nov 30 '22 at 00:51
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    Can you please explain that using calculus and basic measure theory terminology? – mathlander Nov 30 '22 at 00:52
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    Any continuous linear functional $F : C([a,b]) \to \mathbb{R}$ can be represented as integration with respect to a finite Borel measure $\mu$ on $[a,b]$, i.e. $F(\phi) = \int \phi d\mu$. This is what it means for finite Borel measures to be dual to continuous functions. So to uniquely characterize a continuous linear functional, it suffices to uniquely characterize its corresponding measure. – kieransquared Nov 30 '22 at 00:58
  • That makes sense. – mathlander Nov 30 '22 at 00:59
  • It still doesn't answer the question, though. – mathlander Nov 30 '22 at 18:38
  • Related: https://math.stackexchange.com/questions/2801548/is-there-an-axiomatic-characterization-of-the-lebesgue-integral – user76284 Dec 13 '22 at 07:54

1 Answers1

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The Riemann integral is characterized by:

  • $\int_a^b c \; dx = c (b-a)$
  • $\int_a^c f(x) \;dx = \int_a^b f(x) \; dx + \int_b^c f(x) \;dx$

Together these first two properties ensure that the integral of any piecewise constant function (i.e. the area of a bunch of Riemann sum rectangles) is correct.

  • If $f(x) \leq g(x)$ everywhere, then $\int_a^b f(x) \; dx \leq \int_a^b g(x) \; dx$

This property ensures that the integral of a function is between its upper and lower Riemann sums, so if they converge together, the value they converge to must be the integral.

Perhaps this can inspire the kind of characterization you're looking for?

TomKern
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  • The question is about antidifferentiation, not definite integration. – mr_e_man Dec 13 '22 at 01:09
  • This would work, since antidifferentiation and definite integration can easily be defined in terms of each other. – mathlander Dec 13 '22 at 01:27
  • I admit I'm not fully content with this answer: hopefully there's an alternative more along the lines of "the unique linear operator satisfying u-substitution" or something. – TomKern Dec 13 '22 at 13:26