Convergence Theorems - Applications

TheoremLebesgue Differentiation Theorem

Let $f \in L^1_{\text{loc}}(\mathbb{R}^n)$ (locally integrable). For almost every $x \in \mathbb{R}^n$ : $\lim_{r \to 0} \frac{1}{\lambda(B_r(x))} \int_{B_r(x)} f(y) \, dy = f(x)$

where $B_r(x)$ denotes the ball of radius $r$ centered at $x$ , and $\lambda$ is Lebesgue measure.

Points where this limit equals $f(x)$ are called Lebesgue points of $f$ . The theorem states that almost every point is a Lebesgue point.

This profound result shows that for integrable functions, averaging over shrinking balls recovers the function's value at almost every point. It generalizes the Fundamental Theorem of Calculus and connects integration with differentiation.

ExampleApplication to Continuity

If $f$ is continuous at $x$ , then $x$ is automatically a Lebesgue point. The theorem shows that even for discontinuous integrable functions, "most" points behave as if the function were continuous there in an averaged sense.

For instance, consider $f = \chi_{[0,1]}$ on $\mathbb{R}$ :

For $x \in (0, 1)$ : Small balls around $x$ are contained in $[0,1]$ , so the average is $1 = f(x)$
For $x \notin [0, 1]$ : Small balls are disjoint from $[0,1]$ , so the average is $0 = f(x)$
At $x = 0$ or $x = 1$ : The limit exists and equals $1/2$ (half the ball is in $[0,1]$ )

The Lebesgue points are all points in $\mathbb{R}$ .

Remark

Applications and consequences:

Density theorem: The Lebesgue Density Theorem (Chapter 2) is a special case where $f = \chi_E$ for a measurable set $E$ .
Approximate continuity: A function is approximately continuous at $x$ if $x$ is a Lebesgue point. Thus integrable functions are approximately continuous almost everywhere.
Differentiation of integrals: If $F(x) = \int_{-\infty}^x f(t) \, dt$ for $f \in L^1(\mathbb{R})$ , then $F'(x) = f(x)$ for almost every $x$ . This is the Lebesgue version of the Fundamental Theorem of Calculus.
Hardy-Littlewood maximal function: The proof uses the maximal function: $\mathcal{M}f(x) = \sup_{r > 0} \frac{1}{\lambda(B_r(x))} \int_{B_r(x)} |f(y)| \, dy$

The Hardy-Littlewood maximal inequality states that $\mathcal{M}f$ is weak $L^1$ , which controls the size of the exceptional set.

The Lebesgue Differentiation Theorem is central to harmonic analysis, partial differential equations, and geometric measure theory. It reveals that integration and differentiation are inverse operations in a measure-theoretic sense, extending classical calculus to the Lebesgue setting.