Kolmogorov almost-everywhere divergence — recorded theorem

There exists $f\in L^1(\mathbb{T})$, where $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ has Haar mass one, such that

${\sup }_{N\ge 0}\mathrm{\mid }{S}_{N}f(x)\mathrm{\mid }=\mathrm{\infty }\text{for almost every }x\in \mathbb{T}\mathrm{.}$

Equivalently, the function in Carleson maximal partial-sum operator is infinite almost everywhere for this $f$. Every individual partial sum is finite. The Fourier series therefore diverges almost everywhere.

This records Grafakos, Theorem 4.2.1 and the explicit conclusion (4.2.13). No local proof is supplied, and the claim is not strengthened to divergence at every point.

The source forms a summable weighted series of the polynomials in Kolmogorov polynomial blocks — recorded construction lemma ‡. It chooses weights and polynomial degrees recursively. The large contribution of the current polynomial must exceed both the contribution of earlier polynomials and the tail; those two errors require separate estimates. A full-measure limsup of the good sets supplies arbitrarily large partial sums of the final integrable function. The polynomial construction and this summation argument remain external, so the linked record is a bibliographic mention rather than a logical prerequisite.