CantorLebesgue

The Cantor-Lebesgue theorem is a fundamental result in real analysis concerning the behavior of trigonometric series. It states that if a trigonometric series converges to a function $f(x)$ almost everywhere on a set $E$ with positive Lebesgue measure, and if the coefficients of the series tend to zero, then the function $f(x)$ is Lebesgue integrable on $E$. Furthermore, the integral of $f(x)$ over $E$ can be expressed in terms of the Fourier coefficients.

More formally, consider a trigonometric series of the form $\frac{a_0}{2} + \sum_{n=1}^\infty (a_n \cos(nx) + b_n \sin(nx))$. If this series converges to $f(x)$ for almost every $x$ in a set $E$ of positive Lebesgue measure, and if $\lim_{n\to\infty} a_n = 0$ and $\lim_{n\to\infty} b_n = 0$, then the function $f(x)$ is Lebesgue integrable on $E$.

A significant consequence of the Cantor-Lebesgue theorem is that if a trigonometric series converges everywhere to a bounded function, then its coefficients must tend to zero. This is often referred to as the "Riemann-Lebesgue lemma" for Fourier series, although the lemma itself is a more general statement about Fourier transforms. The theorem has implications for understanding the regularity of functions defined by convergent series and is crucial in the study of Fourier analysis and measure theory. It highlights that convergence almost everywhere and vanishing coefficients are strong conditions for integrability.