In this paper, we obtained a new method of generating finite or countably infinite dimensional product σ —algebra, and proved that countably infinite dimensional Borel field has the cardinality of the continuum.

            本文得到了有限及可數(shù)無(wú)限維乘積σ—代數(shù)的一種新的生成方法，并證明了可數(shù)無(wú)限維Ｂｏｒｅｌ域具有連續(xù)統(tǒng)的勢(shì)．

         For metric space X, by M(X) we mean the set of Borel probability measures over X with bounded support, and by R(X) the set of Radon measures in M(X).M(X), R(X) are endowed with Hutchison metric. In this paper, the following results weregot: R(X) is complete if and only if X is complete and bounded. In case that X is seperable,then M(X) is complete if and only if X is complete and bounded.

            距離空間X上，以M（X）記定義在X的Borel域上的具有有界支撐的概率測(cè)度全體，R（X）記M（X）中所有Radon測(cè)度，M（X），R（X）均賦予Hutchison度量．本文得到：R（X）完備當(dāng)且僅當(dāng)X完備且有界．若X可分，則：M（X）完備當(dāng)且僅當(dāng)X完備且有界．

         Let X be a Hausdorff topological space and m be the finite measure on its Borel σ-field B(X). Let {Tt}t≥0 be the sub-Markov semigroup on L~P(X, m) (p > 1) and F_(r,p). be the Sobolev space generated by {Tt}t≥0 Let Cap_(r,p).(.) (r > 0,p > 1) be the capacity associated with {Tt}t≥0 With some conditions we prove that for any positive functional on F_(r.p)~* the dual space of F_(r,p)., there exists an unique measure μ■ on B(X) satisfying Furthermore for any B ∈ B(X), Cap(r,p).(B) = 0 if and only if μ■(B)...

            設(shè)Ｘ是 Ｈａｕｓｄｏｒｆｆ拓?fù)淇臻g，ｍ是其 Ｂｏｒｅｌ域 Ｂ（Ｘ）上的有限測(cè)度．｛Ｔ_ｔ｝ｔ≥０。是 Ｌ~ｐ（Ｘ；ｍ）（ｐ＞１）上的次馬氏半群．Ｆ_（ｒ，，ｐ）。是由該半群生成的Ｓｏｂｏｌｅｖ空間．Ｃａｐ_（ｒ，ｐ）（ｒ＞ ０；ｐ＞１）是相應(yīng)的容度，本文在一定條件下證明了對(duì)任意Ｆ_（ｒ，ｐ）共軛空間Ｆ_（ｒ，ｐ）~＊中的正泛函■， 存在Ｘ上唯一的σ－有限測(cè)度μ■，使得_（Ｆ（ｒ，ｐ））〈ｕ，■〉_（Ｆ（ｒ，ｐ）＊）＝∫_ｘ~ｕ（ｘ）μ■（ｄｘ），ｕ∈Ｆ_（ｒ，ｐ）， 并且對(duì)任意Ｂ∈Ｂ（Ｘ）Ｃａｐ~（ｒ，ｐ）（Ｂ）＝０的充要條件是μ■（Ｂ）＝０，■∈Ｆ_（ｒ，ｐ）~＊。