Measurable Functions - Applications

TheoremLusin's Theorem

Let $f: \mathbb{R} \to \mathbb{R}$ be a Lebesgue measurable function. For every $\epsilon > 0$ , there exists a closed set $F \subseteq \mathbb{R}$ such that:

$\lambda(F^c) < \epsilon$
The restriction $f|_F: F \to \mathbb{R}$ is continuous

Moreover, if $f$ is bounded, we can choose a continuous function $g: \mathbb{R} \to \mathbb{R}$ such that $\lambda(\{x : f(x) \neq g(x)\}) < \epsilon$ and $\|g\|_\infty \leq \|f\|_\infty$ .

Lusin's Theorem is sometimes paraphrased as "measurable functions are nearly continuous." It reveals that measurable functions, despite being potentially very irregular, are continuous when restricted to large sets.

ExampleApplication to Approximation

Let $f = \chi_{\mathbb{Q} \cap [0,1]}$ be the characteristic function of rationals in $[0,1]$ . This function is discontinuous at every point.

However, given $\epsilon > 0$ , Lusin's theorem provides a closed set $F \subseteq [0,1]$ with $\lambda(F^c) < \epsilon$ such that $f|_F$ is continuous.

Since $\lambda(\mathbb{Q} \cap [0,1]) = 0$ , we can take $F = [0,1] \setminus (\mathbb{Q} \cap [0,1]) = [0,1] \cap \mathbb{Q}^c$ , which has measure $1$ . On this set, $f$ is identically $0$ , hence continuous.

Remark

Consequences and applications of Lusin's Theorem:

Density of continuous functions: The set of continuous functions is dense in $L^p$ spaces for $1 \leq p < \infty$ . Given a measurable function $f \in L^p$ , we can approximate it arbitrarily well by continuous functions.
Tietze Extension: Combined with the Tietze Extension Theorem, Lusin's theorem allows us to extend the continuous restriction to a continuous function on all of $\mathbb{R}$ .
Characterization of measurability: A function is measurable if and only if it is "nearly continuous" in the sense of Lusin's theorem.
Bridge to topology: The theorem connects measure theory with topology, showing that measurable functions inherit some continuity properties.

The proof for bounded functions proceeds by first establishing the result for simple functions (which is straightforward), then approximating general measurable functions by simple functions, and using a diagonal argument to control the errors. The key ingredient is the regularity of Lebesgue measure.