Egoroff's Theorem

2011

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Summary.In this article, we define the Riemann integral on functions from R into real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to a wider range of functions. PreliminariesLet X be a real normed space, let A be a closed-interval subset of R, let f be a function from A into the carrier of X, and let D be a Division of A. A finite sequence of elements of X is said to be a middle volume of f and D if it satisfies the conditions (Def. 1).(Def. 1)(i) len it = len D, and (ii) for every natural number i such that i ∈ dom D there exists a point c of X such that c ∈ rng(f divset(D, i)) and it(i) = vol(divset (D, i)) · c. 17

mentioning

confidence: 99%

Riemann Integral of Functions from R into Real Normed Space

Miyajima¹,

Kato²,

2011

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“…The articles [12], [26], [2], [8], [1], [21], [27], [9], [11], [3], [18], [10], [22], [4], [5], [17], [23], [20], [28], [6], [7], [16], [14], [24], [19], [25], [15], and [13] provide the notation and terminology for this paper.…”

mentioning

confidence: 99%

The Measurability of Complex-Valued Functional Sequences

Narita¹,

Endou²,

2009

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Summary.In this article, we formalized the measurability of complexvalued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable functions. In addition, we proved properties of complex-valued simple functions. Real-Valued Functional SequencesFor simplicity, we adopt the following rules: X is a non empty set, Y is a set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to C, r is a real number, k is a real number, and E is an element of S.Let X be a non empty set and let f be a sequence of partial functions from X into R. The functor R(f ) yields a sequence of partial functions from X into R and is defined by: 89

“…The terminology and notation used in this paper have been introduced in the following papers: [23], [24], [6], [2], [25], [8], [7], [1], [4], [3], [5], [20], [10], [14], [12], [13], [18], [22], [19], [26], [9], [11], [15], [17], and [16].…”

mentioning

confidence: 99%

Riemann Integral of Functions from R into n-dimensional Real Normed Space

Miyajima¹,

Korniłowicz²,

2012

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Summary. In this article, we define the Riemann integral on functions Rinto n-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the [21].MML identifier: INTEGR19, version: 7.1 .0 4.1 .11The terminology and notation used in this paper have been introduced in the following papers: [23] On the Functions from R into n-dimensional Real SpaceFor simplicity, we adopt the following convention: X denotes a set, n denotes an element of N, a, b, c, d, e, r, x 0 denote real numbers, A denotes a non empty closed-interval subset of R, f , g, h denote partial functions from R to R n , and E denotes an element of R n . We now state a number of propositions:(