Thawj txoj kev xav Pugh ua pov thawj ib zaug nws txhais lub Riemann Integral yog tias kev sib koom ua ke cuam tshuam txog kev cia siab. Qhov no yog Theorem 15 ntawm nplooj 155 hauv kuv tsab ntawv. Qhov no mus qhia tias ib tug yuav tsum xub pom zoo rau cov lus txhais.
Puas Riemann integrable implies bounded?
Theorem 4. Txhua Riemann integrable muaj nuj nqi yog bounded.
Puas yog cov haujlwm tsis muaj kev sib koom ua ke?
Ib txoj haujlwm tsis muaj kev sib koom ua ke tsis yog Riemann sib koom ua ke. Hauv qab no, "kev sib koom ua ke" yuav txhais tau tias "Riemann integrable, thiab "integral" yuav txhais tau tias "Riemann inte-gral" tshwj tsis yog tau hais meej meej. f(x)={ 1/x yog 0 < x ≤ 1, 0 yog x=0. yog li lub Upper Riemann sums ntawm f tsis zoo-txhais.
Puas yog Lebesgue ua haujlwm sib koom ua ke?
ntsuas kev ua haujlwm uas muaj bounded yog sib npaug rau Lebesgue integrable functions. Yog hais tias f yog ib tug bounded muaj nuj nqi teev nyob rau hauv ib tug ntsuas tau teev E nrog finite ntsuas. Ces f yog ntsuas yog thiab tsuas yog f yog Lebesgue integrable. … Ntawm qhov tod tes, kev ntsuas ntsuas tau yog "yuav luag" txuas ntxiv.
Koj paub li cas yog tias lub luag haujlwm yog Lebesgue sib koom ua ke?
Yog f, g yog cov haujlwm xws li f=g yuav luag txhua qhov chaw, ces f yog Lebesgue integrable yog tias thiab tsuas yog g yog Lebesgue integrable, thiab kev sib xyaw ntawm f thiab g yog tib yam yog lawv muaj.
