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Integration
Published in Hugo D. Junghenn, Principles of Analysis, 2018
As noted in the introduction to Chapter 1, the Lebesgue integral has several distinct advantages over the Riemann integral. First (proper) Riemann integration takes place on compact subintervals of Rd $ \mathbb R ^{d} $ while no such restriction is placed on the Lebesgue integral. Second, the class of functions that are Lebesgue integrable on compact intervals is much larger than the class of Riemann integrable functions. Third, and perhaps most importantly, the Lebesgue theory makes available powerful tools in the form of limit theorems such as the monotone convergence theorem and the dominated convergence theorem, leading to many important results in analysis and its applications. Nevertheless, the Riemann integral still plays an important role in mathematics and the sciences and as such is worthy of discussion here. In this section we give a brief description of the d-dimensional Riemann integral and compare it to the Lebesgue integral.
Infinite Series
Published in John Srdjan Petrovic, Advanced Calculus, 2020
One of the powerful tools to detect the convergence of a sequence was the Monotone Convergence Theorem (Theorem 2.4.7). We can apply this result to the sequence {sn} if it is monotone increasing, i.e. sn+1 ≤ sn , for all n ∈ ℕ. This condition implies that an = sn+1−sn ≤ 0, so it is natural to consider, for the moment at least, those series that have non-negative terms. From the Monotone Convergence Theorem we immediately obtain the following result.
Solution properties of convex sweeping processes with velocity constraints
Published in Applicable Analysis, 2023
Thus, for every . Since is measurable for all , the function is also measurable for all . As is an increasing sequence of real-valued functions, by the monotone convergence theorem [19, Theorem 4.1] one can assert that converges to a function almost everywhere on . Since for all , we see that . On the other hand, for , we have