Measure Theory & Lebesgue Integration

The Riemann integral, while foundational for introductory calculus, fails for functions that are “too discontinuous.” Measure theory provides a more robust framework by partitioning the range of a function rather than its domain.

1. The Failure of Riemann Integration

Consider the Dirichlet function, which is $1$ for rationals and $0$ for irrationals: $\mathbf{1}_{\mathbb{Q}}(x) = \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q} \end{cases}$

In Riemann integration, any sub-interval contains both a rational and an irrational number. Thus, the lower rectangles always have height $0$ and the upper rectangles height $1$. They never meet.

The Lebesgue approach asks: “How ‘big’ is the set of rationals vs the set of irrationals?“. Since the rationals are countable, they have measure zero. The integral is effectively $0 \cdot (\text{size of irrationals}) + 1 \cdot (\text{size of rationals}) = 0$.

2. Measurable Spaces and $\sigma$-Algebras

To measure a set, we must first decide which sets are “legal” to measure.

Definition: $\sigma$-Algebra

A collection $\mathcal{F}$ of subsets of $X$ is a $\sigma$-algebra if:

1. $X \in \mathcal{F}$
2. $A \in \mathcal{F} \implies A^c \in \mathcal{F}$ (Closed under complements)
3. $A_n \in \mathcal{F} \implies \bigcup_{n=1}^\infty A_n \in \mathcal{F}$ (Closed under countable unions)

Definition: Measure

A measure $\mu$ is a function $\mu: \mathcal{F} \to [0, \infty]$ such that:

1. $\mu(\emptyset) = 0$
2. Countable Additivity: For disjoint sets $A_i$, $\mu(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty \mu(A_i)$.

3. Measurable Functions

A function $f: X \to \mathbb{R}$ is measurable if for every Borel set $B \in \mathcal{B}(\mathbb{R})$, the preimage $f^{-1}(B)$ is in $\mathcal{F}$. Equivalently: $\{x \in X : f(x) > a \} \in \mathcal{F} \quad \forall a \in \mathbb{R}$

import numpy as np
import matplotlib.pyplot as plt

# A "discontinuous" function: 1 if sin(x) > 0, else 0
x = np.linspace(0, 10, 1000)
y = (np.sin(x) > 0).astype(float)

# In Lebesgue terms, we are measuring the set {x : f(x) = 1}
# This set is a collection of intervals [0, pi], [2pi, 3pi], etc.
# These intervals are easy to measure!

plt.figure(figsize=(10, 4))
plt.plot(x, y, label="f(x)")
plt.fill_between(x, y, alpha=0.3)
plt.title("Step Function (Easily Lebesgue Integrable)")
plt.legend()
plt.show()

4. The Lebesgue Integral Construction

The Lebesgue integral is built in three stages:

1. Simple Function: $s(x) = \sum_{i=1}^n a_i \mathbf{1}_{E_i}(x)$, where $E_i$ are measurable disjoint sets.
2. Integral of Simple Function: $\int_X s \, d\mu = \sum_{i=1}^n a_i \mu(E_i)$.
3. General Integrable Function: For a non-negative measurable function $f$: $\int_X f \, d\mu = \sup \left\{ \int_X s \, d\mu : 0 \le s \le f, s \text{ is simple} \right\}$

For general $f$, we split it into $f^+ = \max(f, 0)$ and $f^- = \max(-f, 0)$ and compute $\int f^+ d\mu - \int f^- d\mu$.

5. Almost Everywhere (a.e.)

In measure theory, we don’t care about what happens on “tiny” sets of measure zero. We say a property holds almost everywhere if the set of points where it fails has measure zero.

Example: If $f(x) = g(x)$ for all $x$ except at $x=0$, their Lebesgue integrals are identical. This is why we can integrate functions with “holes” or infinite spikes (as long as the spikes aren’t too “fat”).

6. Convergence Theorems

The true power of Lebesgue theory is how well it handles limits.

• Monotone Convergence: If $f_n$ grows towards $f$, the integrals grow towards the integral of $f$.
• Dominated Convergence: If $f_n \to f$ and all $f_n$ stay under some “umbrella” function $g$ that is integrable, then the limit of the integrals is the integral of the limit.