Riemann Integral of Functions from R into R n

Transcription

1 FORMLIZED MTHEMTICS Vol. 17, No. 2, Pges , 2009 DOI: /v y Riemnn Integrl of Functions from R into R n Keiichi Miyjim Ibrki University Hitchi, Jpn Ysunri Shidm Shinshu University Ngno, Jpn Summry. In this rticle, we define the Riemnn Integrl of functions from R into R n, nd prove the linerity of this opertor. The presented method is bsed on [21]. MML identifier: INTEGR15, version: The rticles [22], [1], [23], [5], [6], [15], [20], [24], [7], [17], [16], [2], [4], [3], [8], [18], [9], [12], [10], [14], [13], [19], nd [11] provide the nottion nd terminology for this pper. 1. Preliminries Let be closed-intervl subset of R, let f be function from into R, let S be non empty Division of, nd let D be n element of S. finite sequence of elements of R is sid to be middle volume of f nd D if it stisfies the conditions (Def. 1). (Def. 1)(i) len it = len D, nd (ii) for every nturl number i such tht i dom D there exists n element r of R such tht r rng(f divset(d, i)) nd it(i) = r vol(divset(d, i)). Let be closed-intervl subset of R, let f be function from into R, let S be non empty Division of, let D be n element of S, nd let F be middle volume of f nd D. The functor middle sum(f, F ) yielding rel number is defined s follows: (Def. 2) middle sum(f, F ) = F. We now stte four propositions: 175 c 2009 University of Biłystok ISSN (p), (e)

2 176 keiichi miyjim nd ysunri shidm (1) Let be closed-intervl subset of R, f be function from into R, S be non empty Division of, D be n element of S, nd F be middle volume of f nd D. If f is lower bounded, then lower sum(f, D) middle sum(f, F ). (2) Let be closed-intervl subset of R, f be function from into R, S be non empty Division of, D be n element of S, nd F be middle volume of f nd D. If f is upper bounded, then middle sum(f, F ) upper sum(f, D). (3) Let be closed-intervl subset of R, f be function from into R, S be non empty Division of, D be n element of S, nd e be rel number. Suppose f is lower bounded nd 0 < e. Then there exists middle volume F of f nd D such tht middle sum(f, F ) lower sum(f, D) + e. (4) Let be closed-intervl subset of R, f be function from into R, S be non empty Division of, D be n element of S, nd e be rel number. Suppose f is upper bounded nd 0 < e. Then there exists middle volume F of f nd D such tht upper sum(f, D) e middle sum(f, F ). Let be closed-intervl subset of R, let f be function from into R, nd let T be DivSequence of. function from N into R is sid to be middle volume sequence of f nd T if: (Def. 3) For every element k of N holds it(k) is middle volume of f nd T (k). Let be closed-intervl subset of R, let f be function from into R, let T be DivSequence of, let S be middle volume sequence of f nd T, nd let k be n element of N. Then S(k) is middle volume of f nd T (k). Let be closed-intervl subset of R, let f be function from into R, let T be DivSequence of, nd let S be middle volume sequence of f nd T. The functor middle sum(f, S) yields sequence of rel numbers nd is defined s follows: (Def. 4) For every element i of N holds (middle sum(f, S))(i) = middle sum(f, S(i)). One cn prove the following propositions: (5) Let be closed-intervl subset of R, f be function from into R, T be DivSequence of, S be middle volume sequence of f nd T, nd i be n element of N. If f is lower bounded, then (lower sum(f, T ))(i) (middle sum(f, S))(i). (6) Let be closed-intervl subset of R, f be function from into R, T be DivSequence of, S be middle volume sequence of f nd T, nd i be n element of N. If f is upper bounded, then (middle sum(f, S))(i) (upper sum(f, T ))(i). (7) Let be closed-intervl subset of R, f be function from into R, T be DivSequence of, nd e be n element of R. Suppose 0 < e nd f is lower bounded. Then there exists middle volume sequence S of

3 riemnn integrl of functions from f nd T such tht for every element i of N holds (middle sum(f, S))(i) (lower sum(f, T ))(i) + e. (8) Let be closed-intervl subset of R, f be function from into R, T be DivSequence of, nd e be n element of R. Suppose 0 < e nd f is upper bounded. Then there exists middle volume sequence S of f nd T such tht for every element i of N holds (upper sum(f, T ))(i) e (middle sum(f, S))(i). (9) Let be closed-intervl subset of R, f be function from into R, T be DivSequence of, nd S be middle volume sequence of f nd T. Suppose f is bounded nd f is integrble on nd δ T is convergent nd lim(δ T ) = 0. Then middle sum(f, S) is convergent nd lim middle sum(f, S) = integrl f. (10) Let be closed-intervl subset of R nd f be function from into R. Suppose f is bounded. Then f is integrble on if nd only if there exists rel number I such tht for every DivSequence T of nd for every middle volume sequence S of f nd T such tht δ T is convergent nd lim(δ T ) = 0 holds middle sum(f, S) is convergent nd lim middle sum(f, S) = I. Let n be n element of N, let be closed-intervl subset of R, let f be function from into R n, let S be non empty Division of, nd let D be n element of S. finite sequence of elements of R n is sid to be middle volume of f nd D if it stisfies the conditions (Def. 5). (Def. 5)(i) len it = len D, nd (ii) for every nturl number i such tht i dom D there exists n element r of R n such tht r rng(f divset(d, i)) nd it(i) = vol(divset(d, i)) r. Let n be n element of N, let be closed-intervl subset of R, let f be function from into R n, let S be non empty Division of, let D be n element of S, nd let F be middle volume of f nd D. The functor middle sum(f, F ) yields n element of R n nd is defined by the condition (Def. 6). (Def. 6) Let i be n element of N. Suppose i Seg n. Then there exists finite sequence F 1 of elements of R such tht F 1 = proj(i, n) F nd (middle sum(f, F ))(i) = F 1. Let n be n element of N, let be closed-intervl subset of R, let f be function from into R n, nd let T be DivSequence of. function from N into (R n ) is sid to be middle volume sequence of f nd T if: (Def. 7) For every element k of N holds it(k) is middle volume of f nd T (k). Let n be n element of N, let be closed-intervl subset of R, let f be function from into R n, let T be DivSequence of, let S be middle volume sequence of f nd T, nd let k be n element of N. Then S(k) is middle volume of f nd T (k). Let n be n element of N, let be closed-intervl subset of R, let f be

4 178 keiichi miyjim nd ysunri shidm function from into R n, let T be DivSequence of, nd let S be middle volume sequence of f nd T. The functor middle sum(f, S) yields sequence of E n, nd is defined by: (Def. 8) For every element i of N holds (middle sum(f, S))(i) = middle sum(f, S(i)). Let n be n element of N, let Z be non empty set, nd let f, g be prtil functions from Z to R n. The functor f + g yielding prtil function from Z to R n is defined by: (Def. 9) dom(f + g) = dom f dom g nd for every element c of Z such tht c dom(f + g) holds (f + g) c = f c + g c. The functor f g yields prtil function from Z to R n nd is defined by: (Def. 10) dom(f g) = dom f dom g nd for every element c of Z such tht c dom(f g) holds (f g) c = f c g c. Let n be n element of N, let r be rel number, let Z be non empty set, nd let f be prtil function from Z to R n. The functor r f yields prtil function from Z to R n nd is defined s follows: (Def. 11) dom(r f) = dom f nd for every element c of Z such tht c dom(r f) holds (r f) c = r f c. 2. Definition of Riemnn Integrl of Functions from R into R n Let n be n element of N, let be closed-intervl subset of R, nd let f be function from into R n. We sy tht f is bounded if nd only if: (Def. 12) For every element i of N such tht i Seg n holds proj(i, n) f is bounded. Let n be n element of N, let be closed-intervl subset of R, nd let f be function from into R n. We sy tht f is integrble if nd only if: (Def. 13) For every element i of N such tht i Seg n holds proj(i, n) f is integrble on. Let n be n element of N, let be closed-intervl subset of R, nd let f be function from into R n. The functor integrl f yields n element of R n nd is defined by: (Def. 14) dom integrl f = Seg n nd for every element i of N such tht i Seg n holds (integrl f)(i) = integrl proj(i, n) f. The following propositions re true: (11) Let n be n element of N, be closed-intervl subset of R, f be function from into R n, T be DivSequence of, nd S be middle volume sequence of f nd T. Suppose f is bounded nd integrble nd δ T is convergent nd lim(δ T ) = 0. Then middle sum(f, S) is convergent nd lim middle sum(f, S) = integrl f.

5 riemnn integrl of functions from (12) Let n be n element of N, be closed-intervl subset of R, nd f be function from into R n. Suppose f is bounded. Then f is integrble if nd only if there exists n element I of R n such tht for every DivSequence T of nd for every middle volume sequence S of f nd T such tht δ T is convergent nd lim(δ T ) = 0 holds middle sum(f, S) is convergent nd lim middle sum(f, S) = I. Let n be n element of N nd let f be prtil function from R to R n. We sy tht f is bounded if nd only if: (Def. 15) For every element i of N such tht i Seg n holds proj(i, n) f is bounded. Let n be n element of N, let be closed-intervl subset of R, nd let f be prtil function from R to R n. We sy tht f is integrble on if nd only if: (Def. 16) For every element i of N such tht i Seg n holds proj(i, n) f is integrble on. Let n be n element of N, let be closed-intervl subset of R, nd let f be prtil function from R to R n. The functor f(x)dx yielding n element of R n is defined by: (Def. 17) dom f(x)dx = Seg n nd for every element i of N such tht i Seg n holds ( f(x)dx)(i) = (proj(i, n) f)(x)dx. One cn prove the following two propositions: (13) Let n be n element of N, be closed-intervl subset of R, f be prtil function from R to R n, nd g be function from into R n. Suppose f = g. Then f is integrble on if nd only if g is integrble. (14) Let n be n element of N, be closed-intervl subset of R, f be prtil function from R to R n, nd g be function from into R n. If f = g, then f(x)dx = integrl g. Let, b be rel numbers, let n be n element of N, nd let f be prtil function from R to R n. The functor b f(x)dx yields n element of R n nd is defined s follows: b (Def. 18) dom f(x)dx = Seg n nd for every element i of N such tht i Seg n holds ( b f(x)dx)(i) = b (proj(i, n) f)(x)dx.

6 180 keiichi miyjim nd ysunri shidm 3. Linerity of Integrtion Opertor The following propositions re true: (15) Let n be n element of N, f 1, f 2 be prtil functions from R to R n, nd i be n element of N. If i Seg n, then proj(i, n) (f 1 + f 2 ) = proj(i, n) f 1 + proj(i, n) f 2 nd proj(i, n) (f 1 f 2 ) = proj(i, n) f 1 proj(i, n) f 2. (16) Let n be n element of N, r be rel number, f be prtil function from R to R n, nd i be n element of N. If i Seg n, then proj(i, n) (r f) = r (proj(i, n) f). (17) Let n be n element of N, be closed-intervl subset of R, nd f 1, f 2 be prtil functions from R to R n. Suppose f 1 is integrble on nd f 2 is integrble on nd dom f 1 nd dom f 2 nd f 1 is bounded nd f 2 is bounded. Then f 1 + f 2 is integrble on nd f 1 f 2 is integrble on nd (f 1 + f 2 )(x)dx = f 1 (x)dx + f 2 (x)dx nd (f 1 f 2 )(x)dx = f 1 (x)dx f 2 (x)dx. (18) Let n be n element of N, r be rel number, be closed-intervl subset of R, nd f be prtil function from R to R n. Suppose dom f nd f is integrble on nd f is bounded. Then r f is integrble on nd (r f)(x)dx = r f(x)dx. (19) Let n be n element of N, f be prtil function from R to R n, be closed-intervl subset of R, nd, b be rel numbers. If = [, b], then b f(x)dx = f(x)dx. (20) Let n be n element of N, f be prtil function from R to R n, be closed-intervl subset of R, nd, b be rel numbers. If = [b, ], then b f(x)dx = f(x)dx. References [1] Grzegorz Bncerek. The ordinl numbers. Formlized Mthemtics, 1(1):91 96, [2] Grzegorz Bncerek nd Krzysztof Hryniewiecki. Segments of nturl numbers nd finite sequences. Formlized Mthemtics, 1(1): , [3] Czesłw Byliński. Binry opertions pplied to finite sequences. Formlized Mthemtics, 1(4): , [4] Czesłw Byliński. Finite sequences nd tuples of elements of non-empty sets. Formlized Mthemtics, 1(3): , [5] Czesłw Byliński. Functions nd their bsic properties. Formlized Mthemtics, 1(1):55 65, 1990.

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