Integrating with respect to Lebesgue measure

Question

I have taken a course on measure theory, learned all the important limit theorems, construction of the Lebesgue integral and so forth, but what I have never encountered is, how I actually compute a Lebesgue integral w.r.t. a specific given measure. I.e., how the definition of the measure with respect to which we integrate comes into play when computing the integral. On $\mathbb{R}$, for example, consider the Borel Lebesgue measure $\lambda((a, b])=b - a$ or the counting measure $\nu(A)=|A|$ (if $A$ is a measurable set). How do you compute the integral of a function $f$ with respect to these measures and where and how do their definitions come into play?

I am aware that on compact intervals integrating a function $f$ w.r.t. the Borel-Lebesgue measure is the same as just computing the Riemann Integral, but that is not what I mean. I mean how do you use the specific definition of your measure in the (Lebesgue) integral to compute that integral without resorting to the Riemann Integral (which is anyway only possible if your measure is the Borel-Lebesgue one)?

Also, the Borel-Lebesgue measure is defined for intervals, but how do you compute the integral w.r.t. the Borel-Lebesgue measure for an arbitrary measurable set?

Answer 1

One approach is to approximate the function you want to integrate with a sequence of simple functions, then apply the monotone convergence theorem.

Given a positive measurable function $f$, it is easy to construct an increasing sequence $s_n$ of simple functions that converges pointwise to $f$. For example

• $s_0$ could be $\mathbf 1_{0 \leq f \leq 1}$
• $s_1$ could be $\tfrac 1 2 \times \mathbf 1_{0 \leq f \leq \tfrac 1 2} + \mathbf 1_{\tfrac 1 2 < f \leq 1} + \tfrac 3 2 \times \mathbf 1_{1 < f \leq \tfrac 3 2} + 2 \times \mathbf 1_{\tfrac 3 2 < f \leq 2} $

... and so on.

$\int f = \lim_{n \to \infty} \int s_n$, by the monotone convergence theorem.

The integral of any simple function $s = \sum_i c_i \mathbf 1_{E_i}$ can be expressed directly in terms of the Lebesgue measures of sets $E_i$. As explained above, the integral of any measurable function $f$ can be expressed as the limit of the integrals of simple functions. Thus we've described the integral of $f$ in terms of Lebesgue measures of sets, which is what we want.

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With regards to your second question, first observe that if you know how to compute the Lebesgue measure of an interval, then you also know how to compute the Lebesgue measure of an open set or a closed set. This is because every open set is the disjoint union of a finite or countable collection of open intervals, and every closed set is the complement of an open set.

To compute the Lebesgue measure of an arbitrary measurable set $E$, we can appeal to the regularity of the Lebesgue measure. This tells us that for any $\epsilon > 0$, there exists a closed set $F \subset E$ and an open set $U \supset E$ such that $m(U - F) < \epsilon$.

In practical terms, if you want to apply this, then you can construct a sequence of nested closed sets $F_1 \subset F_2 \subset F_3 \subset \dots$ with all $F_n \subset E$, and a sequence of nested open sets $U_1 \supset U_2 \supset U_3 \supset \dots $ with all $U_n \supset E$, such that $m(U_n - F_n) < \tfrac 1 n$. Then $m(E) = \lim_{n \to \infty} F_n = \lim_{n \to \infty} U_n$.

Thus the measure of an arbitrary set is the limit of a sequence of measures of open/closed sets, which in turn is given in terms of measures of intervals.