Abstract Measure Grand Lebesgue Spaces and Applications

We introduce abstract measure grand Lebesgue spaces endowed with ball basis structures and investigate their fundamental properties, together with the behavior of $\mathfrak{BO}$ operators. By exploiting sparse domination for $\mathfrak{BO}$ operators and norm inequalities for sparse operators, we derive norm estimates for this class of operators. As applications, we establish that $\mathfrak{BO}$ operators encompass the maximal operators, Calder\'on-Zygmund operators on homogeneous spaces and Carleson operators.