Kolmogorov gliding hump series converges in lone

Apply F1 for each $j\ge 1$ with $H=j{2}^j$ and $\eta =2^{-j}$. AC selects one polynomial from each nonempty set of possible polynomials. Its degree can be taken as its greatest nonzero coefficient index, which exists because its norm is one. The stated exceptional sets are measurable because the maxima are finite continuous maxima. The recursion for $m_j$ is explicit; it gives $m_{j+1}>m_j+d_j$ and $m_j\ge j\ge 1$. F2 therefore places each $Q_j$ in the claimed disjoint positive interval and gives $\mathrm{\parallel }{Q}_j{\mathrm{\parallel }}_1=a_j=2^{-j}$.

For partial sums $f_J={\sum }_{j\le J}{Q}_j$ and $K>J$, the triangle inequality gives $\mathrm{\parallel }{f}_K-f_J{\mathrm{\parallel }}_1\le {\sum }_{j=J+1}^K{2}^{-j}\le 2^{-J}$. They are Cauchy, so F3 supplies an $L^1$ limit f. Passing K to infinity in the norm inequality gives $\mathrm{\parallel }f-f_J{\mathrm{\parallel }}_1\le 2^{-J}$, and $\mathrm{\parallel }f{\mathrm{\parallel }}_1\le {\sum }_{j\ge 1}{2}^{-j}=1$. AC includes the countable choice required by F3.

Apply F4 to $G_J={\sum }_{j\le J}\mathrm{\mid }{Q}_j\mathrm{\mid }$. Its increasing limit G is measurable and has integral $\int G={\lim }_J{\sum }_{j\le J}{2}^{-j}=1$. For every positive integer M, $Mm\{G=\mathrm{\infty }\}\le \int G=1$, so $G<\mathrm{\infty }$ off a null set. There the numerical series converges absolutely; set its sum h to zero on that measurable exceptional set. The resulting function is measurable, $\mathrm{\mid }h\mathrm{\mid }\le G$ a.e., and $\mathrm{\mid }h-f_J\mathrm{\mid }\le {\sum }_{j>J}\mathrm{\mid }{Q}_j\mathrm{\mid }$ a.e. A further application of F4 to the nonnegative tail gives $\mathrm{\parallel }h-f_J{\mathrm{\parallel }}_1\le 2^{-J}$. Consequently $\mathrm{\parallel }h-f{\mathrm{\parallel }}_1\le \mathrm{\parallel }h-f_J{\mathrm{\parallel }}_1+\mathrm{\parallel }{f}_J-f{\mathrm{\parallel }}_1\to 0$, so h represents f. No smallness of completed blocks is attributed to their frequency shifts.