Lebesgue vs. Riemann: Partitioning Strategies

Hello! Welcome to the fourth lesson in our course on the measure-theoretic foundations of probability.

Introduction

In our last lesson, we established the formal definition of a random variable as a measurable function. We saw that this was not just a technicality, but a crucial property ensuring that for a random variable $X$, the set of outcomes $\{\omega \in \Omega \mid X(\omega )\in B\}$ is a valid event to which we can assign a probability, for any "reasonable" set of values $B$.

We motivated this idea by alluding to a different way of thinking about integration—one that involves partitioning the output values of a function, rather than its input values. Today, we will explore this idea in full.

This lesson directly addresses the learning outcome: Contrast the Lebesgue integral with the Riemann integral conceptually, focusing on the difference between partitioning the domain versus the codomain. You'll see how the measure-theoretic machinery we've been building (σ-algebras, measures, measurable functions) comes together to create a more powerful and flexible theory of integration, which is the basis for the modern definition of expectation.

1. The Riemann Integral: A Familiar Approach and Its Limits

From your background in mathematics, you are well-acquainted with the Riemann integral, typically taught as the "area under the curve." The fundamental strategy is to approximate this area using rectangles.

The key idea of the Riemann integral is to partition the domain (the x-axis) into many small subintervals. For each subinterval, we form a rectangle whose height is given by the function's value at some point in that subinterval. The integral is then the limit of the sum of these rectangular areas as the width of the subintervals goes to zero.

While powerful, this approach has significant limitations, which ultimately motivated the search for a new theory of integration.

• Difficulty with Discontinuities: The Riemann integral struggles with functions that are "very" discontinuous. A classic example is the Dirichlet function, which is 1 for rational numbers and 0 for irrational numbers. On any subinterval of the domain, no matter how small, the function's value jumps between 0 and 1. This makes it impossible to define the height of the approximating rectangles in a way that converges to a single value.
• Poor Convergence Properties: In statistics and econometrics, we often need to analyze the limit of a sequence of functions, for example, when proving consistency of an estimator. A crucial question is whether we can interchange the limit and the integral:
  ${\lim }_{n\to \infty }\int f_n(x)dx\stackrel{?}{=}\int \left({\lim }_{n\to \infty }{f}_n(x)\right)dx$
  The conditions under which this is allowed for Riemann integrals (e.g., uniform convergence) are very strict and often not met in practice.
• Lack of Generality: The construction is tied to the structure of the real line (or ${\mathbb{R}}^n$). It is not easily generalized to integration over more abstract spaces, which is precisely what we need to do in probability theory when we want to integrate over a sample space $\Omega$.

2. The Lebesgue Integral: A Conceptual Shift

The genius of Henri Lebesgue was to turn the problem on its head. Instead of partitioning the domain, he proposed to partition the codomain (the y-axis).

This conceptual shift is best understood with an analogy, which Lebesgue himself reportedly used:

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral.

But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.

The Riemann approach takes slices of the domain and asks "how high is the function here?". The Lebesgue approach takes slices of the range and asks "for which x-values does the function have this height?".

Let's formalize this intuition. The Lebesgue integral is constructed as follows:

1. Partition the codomain: Choose a partition of the y-axis, creating a set of horizontal "slabs" of a certain height.
2. Find the preimages: For each slab, find the set of all points on the x-axis whose function values fall within that slab. This set is the preimage of that y-interval.
3. Measure the preimages: This is the critical step where measure theory enters. The preimage might be a complicated, disconnected set (as seen in the image above). We use a measure (like the Lebesgue measure we discussed in Lesson 1.2, which generalizes 'length') to find the "size" of this set.
4. Sum the results: The contribution to the integral from each slab is its height multiplied by the measure of its corresponding preimage. The total integral is the limit of this sum as the slabs get thinner.

You can now see why we needed the concept of a measurable function. For this procedure to work, the preimage of each horizontal slab (which is an interval, a simple Borel set) must be a set whose measure we can actually compute—it must be a measurable set. This is exactly the condition we imposed on random variables in the last lesson.

3. The Payoff: Why the Lebesgue Approach is Superior

This new approach isn't just a different way to get the same answer; it fundamentally expands the class of functions we can integrate and provides a more robust theoretical framework.

Integrating the "Unintegrable"

Let's revisit the Dirichlet function $f(x)$, which is 1 on rationals and 0 on irrationals. The Lebesgue integral handles it with ease.

As the video explains, the calculation is straightforward:

• Partition the codomain: The only values are {0, 1}.
• Value 1: The preimage is the set of rational numbers, $\mathbb{Q}$. The Lebesgue measure of this countable set is 0. So, the contribution is $1\times \mu (\mathbb{Q})=1\times 0=0$.
• Value 0: The preimage is the set of irrational numbers. On an interval like $[0,1]$, this set has a measure of 1. The contribution is $0\times \mu (\mathbb{R}\setminus \mathbb{Q})=0\times 1=0$.
• Total Integral: The Lebesgue integral is $0+0=0$.

This result aligns perfectly with our intuition from probability: if you pick a number at random from $[0,1]$, the probability of picking a rational number is zero.

Generality and Better Convergence

The Lebesgue integral resolves the key limitations of the Riemann integral.

The key advantages are:

1. Generality: The construction depends only on a measure space, not the specific geometry of ${\mathbb{R}}^n$. This allows us to define the integral of a random variable $X$ over an abstract sample space $\Omega$ with respect to a probability measure $P$. This integral is the expected value of $X$:
   $\mathbb{E}[X]={\int }_{\Omega }X(\omega )dP(\omega )$
2. Completeness and Convergence: The space of Lebesgue-integrable functions is complete. This is a deep and powerful property, analogous to how the real numbers "complete" the rational numbers. It ensures that the limits of sequences of integrable functions are themselves integrable. This leads to powerful convergence theorems that allow us to interchange limits and integrals under much weaker conditions than for the Riemann integral.

Conclusion

Let's summarize the fundamental contrast you were asked to learn.

• The Riemann integral partitions the domain (x-axis). It asks, "For this small interval of $x$, what is the value of $f(x)$?" This method fails for highly discontinuous functions and has restrictive convergence properties.
• The Lebesgue integral partitions the codomain (y-axis). It asks, "For this small range of values $y$, what is the measure of the set of $x$'s for which $f(x)$ is in this range?" This method requires the machinery of measure theory but is far more general and powerful.

You now understand the conceptual foundation of the Lebesgue integral and why it is the basis for modern probability theory. The expectation you have used countless times in statistics and econometrics, $\mathbb{E}[X]$, is formally defined as the Lebesgue integral of the random variable $X$ with respect to the underlying probability measure $P$.

Preview of the Next Lesson:
We will explore the most important practical "payoff" of the Lebesgue integral for statistics: the Dominated Convergence Theorem. This theorem provides a remarkably useful and general condition under which we can justify swapping limits and expectations, a step that is fundamental to the proof of many core results in statistical and econometric theory, such as the consistency of estimators.