Given a nonnegative monotone set function and a nonnegative measurable function on a measurable space, the Choquet integral determines a new nonnegative monotone set function that is absolutely continuous with respect to the original one (in a generalized sense for monotone set functions). This new...
2.Continuity of Monotone Set Function and Convergences of Measurable Function Sequence;单调集函数的连续性与可测函数序列的收敛 3.And the sufficient and necessary conditions of the monotone set function which is continuous of the empty set and the full set are given.本文利用单调集函数的4种连续性的...
(iii) U is lower semicontinuous in the weak topology σ(Cb,ca). (iv) There exist a nonempty set M⊆ca+ and a function α:M→R such that U(f)=supμ∈M(∫Ωfdμ−α(μ))for all f∈Cb. (2.2) We apply Corollary 2.5 to the case of Choquet integrals. In the sequel, le...
“The S-N curve for a pure metal will be amonotonicfunction with N increasing as stress decreases.” “How frightful those white-shuttered brick piles whichmonotonizethe streets of Philadelphia!” “From time to time, Ace will, in a jerksome way,monotonizethe conversation with witticisms too...
for every measurable strictly positive function f on (0,∞). Remark 1.2 The proof of Theorem 1.1 will be given in Sect. 2, and it is organized as follows. We will first establish a cobweb of implications pouring down from (i) to (iii) through each of the remaining ones, more precisel...
Maximum Lebesgue Extension 7 It always holds L∞ ⊂ MΦ ⊂ LΦ ⊂ L1 and MΦ = LΦ if Φ satisfies the so-called ∆2-condition, while if for example Φ(x) = e|x| − 1 and (Ω, F , P) is atomless, then L∞ MΦ LΦ L1. The function ρΦ is Lebesgue on MΦ...
LetXXbe a measure space with a positive measureμμand letf:X→[0,∞]f:X→[0,∞]be a measurable function. Prove thatlimn→∞∫Xnlog(1+fn)dμ=∫Xfdμ.limn→∞∫Xnlog(1+fn)dμ=∫Xfdμ. Proof f n loglog ff f1≤f2≤⋯f1≤f2≤⋯ ...
Let (S, µ) be an arbitrary probability space and let W : S ×S → [0, 1] be a symmetric measurable function. (W is sometimes called graphon [7; 8], we will use the alternative kernel denomination =-=[4]-=-.) Let X1, X2, . . . , be an i.i.d. sequence of random ...
Continuity of Monotone Set Function and Convergences of Measurable Function Sequence; 单调集函数的连续性与可测函数序列的收敛 3. And the sufficient and necessary conditions of the monotone set function which is continuous of the empty set and the full set are given. 本文利用单调集函数的4种连续...
Continuity of Monotone Set Function and Convergences of Measurable Function Sequence; 单调集函数的连续性与可测函数序列的收敛 3. And the sufficient and necessary conditions of themonotone set functionwhich is continuous of the empty set and the full set are given. ...