Almost subadditive weight functions form Braun–Meise–Taylor theory of ultradistributions

As it is known, Roumieu–Komatsu theory of ultradistributions is strictly larger than Beurling–Björck one and that the latter theory is established by the class of all subadditive weight functions. In its own turn, Roumieu–Komatsu theory is equivalent to Braun–Meise–Taylor one which is given by the class of all weight functions. We prove that the class of all almost subadditive weight functions forms Braun–Meise–Taylor theory of ultradistributions.