I-Convergence of Sequences of p-Bochner, p-Dunford and p-Pettis Integrable Functions with Values in a Banach Space

We study $\mathcal{I}$-convergence of sequences of functions. Mainly, we study Vitali type $\mathcal{I}$-convergence theorems for sequences of p-Bochner, p-Dunford and p-Pettis integrable functions with values in a Banach space. With the help of these theorems, we characterize p-Bochner relatively compact subsets of p-Bochner integrable functions and p-Pettis relatively $\mathcal{I}$-sequentially compact subsets of p-Dunford integrable functions. We study uniform $\mathcal{I}$-convergence of sequences of p-Bochner, p-Dunford and p-Pettis integrable functions, and weak uniform $\mathcal{I}$-convergence of sequences of p-Dunford and p-Pettis integrable functions, and discuss how they are related to the p-Bochner and p-Pettis $\mathcal{I}$-convergence. $\mathcal{I}$-convergence of compact mappings are studied with special emphasis to compact linear operators on normed linear spaces. We also study $\mathcal{I}$-exhaustiveness, $\mathcal{I}$-weak exhaustiveness and $\mathcal{I}$-$\alpha$-convergence of sequences of metric space-valued functions defined on a metric space, and sequences of p-Dunford integrable functions are discussed in this perspective.