Integration

The concept of an integral on a general measure space is developed from first principles. Riemann–Stieltjes and Lebesgue–Stieltjes integrals are defined. The monotone convergence theorem, fundamental properties of integrals, and related inequalities are covered. Other topics include product measure and multiple integrals, Fubini’s theorem, signed measures, and the Radon–Nikodym theorem.

On Fuzzy Henstock-Kurzweil-Stieltjes-♦-Double Integral on Time scales

We introduce and study some properties of fuzzy Henstock-Kurzweil-Stietljes-$ \Diamond $-double integral on time scales. Also, we state and prove the uniform convergence theorem, monotone convergence theorem and dominated convergence theorem for the fuzzy Henstock-Kurzweil Stieltjes-$\Diamond$-double integrable functions on time scales.

Extended Real-Valued Double Sequence and Its Convergence

In this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.

(G) Fuzzy Integral on Fuzzy Set

In this paper, the concept of (G) fuzzy integral on a fuzzy set is given and the basic properties of it are discussed. Monotone Convergence Theorem and Fatou Lemma are proved. Finally, A necessary and sufficient condition on which (G) fuzzy integrals of two fuzzy measurable functions on fuzzy sets are always equal is given.

A new proof of the monotone convergence theorem of Lebesgue integral on σ-class

In this short note a new proof of the monotone convergence theorem of Lebesgue integral on σ-class is given.