Real Analysis – What Does $dx$ Mean in a Lebesgue Integral?

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This is an introduction for Lebesgue integral of simple function in Carothers' Real Analysis.

We say that a simple function $\phi$ is Lebesgue integrable if the set {$\phi$ $\ne$ 0} has finite measure. In this case, we may write the standard representation for $\phi$ as $\phi = \sum_{i=0}^{n} a_i \chi_{A_i}$, where $a_0 = 0, a_1, .., a_n$ are distinct real numbers, where $A_0 = \{\phi = 0\}, A_1, …, A_n$ are pairwise disjoint and measurable, and where only $A_0$ has infinite measure, Once $\phi$ is so written, there is an obvious definition for $\int \phi$, namely, $$\int \phi = \int_{\mathbb R} \phi = \int_{-\infty}^{+\infty} \phi(x) dx = \sum_{i=1}^{n} a_i m(A_i)$$.

I've noticed that wikipedia's definition of Lebesgue integral(see here https://en.wikipedia.org/wiki/Lebesgue_integration) uses $d\mu$. So What does $dx$ or $d\mu$ mean in Lebesgue integral?

Update:

I don't think it is a exactly duplicate one coz I didn't mean using $d\mu$ instead of $dx$. Before my typing this question, I have read Rodyen's Real Analysis, 3rd and he also uses $dx$ in Lebesgue integral as well. $d\mu$ is just from wikipedia. I have this question in this May when I was reading Caorthers' book and during that time, I treated it as a whole of symbols and being equal to a fixed formula — $\sum_{i=1}^{n}a_i m(A_i)$. And then when I was trying to solve some problems with this symbol in Lebesgue integral, I felt weird for quite a while, recalling the Riemann's definition and then realized "ohhh, man, it is not Riemann integral".

Best Answer

The meaning of the $dx$ in a Lebesgue integral, or any other integral, depends on the extended framework in which the theory is situated. If you are working in a framework in which the measure is the main focus, where there is a wide class of possible measures, for example on a differentiable manifold or in a topological vector spaces, or something more exotic, then the $dx$ would have to be replaced by $d\mu$ to indicate which measure you are talking about. If you're just talking about the standard measure on a Cartesian space $\mathbb{R}^n$, then $dx$ is adequate. The $x$ then serves as a dummy variable (or "bound variable" in mathematical logic) so that you can use inline functions as integrands. For example $\int x^2\,dx$. If the measure is variable, you might like something like $\int x^2\,d\mu(x)$, which indicates the choice of measure and also the dummy variable.

In the framework of differential forms, the $dx$ has a different meaning. But that's a different kettle of fish.

In reality, integration theory has diverged in so many ways, the notations of the 17th and 18th centuries are no longer adequate. The "generalization thrusts" of integration theory include the following.

  1. Generalization from Riemann integrable to very general kinds of functions.
  2. Generalization of the class of measurable sets to a maximal class which extends the Borel measurable sets.
  3. Generalization of the measure from the standard invariant measure on $\mathbb{R}^n$ to axiomatically defined very general measures.
  4. Generalization from real-valued functions to distributions such as Radon measures, Schwartz distributions, Stieltjes integrals, etc.
  5. Generalization from volume measures to surface measures, involving maybe differential forms, for example, as in the Stokes theorem.
  6. Generalization to fractional dimensional measures, like Carathéodory measures, Hausdorff measures etc.
  7. Generalization to extremely discontinuous domain sets, as in geometric measure theory.
  8. Generalization to infinite-dimensional spaces, and spaces which are not vector spaces.

In these generalized frameworks, the traditional notations for integrals look more and more and more inadequate to describe what the integral is. So what I'm saying is that the problem is much bigger than you describe. The integral notation is just a shorthand, and you have to fill in the details by reading the context. I.e all of the notations are wrong or inaccurate or incomplete. And the extent of the inaccuracy or ambiguity is greater in the more generalized frameworks.

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