Differentiation of measures on an arbitrary measurable space
Differentiation of measures on an arbitrary measurable space
复制标题
任意可测空间上的测度微分
DOI：
10.1016/j.jmaa.2023.127438
复制
发表时间：
2023
影响因子：
1.3
通讯作者：
Siorpaes, Pietro
中科院分区：
文献类型：
--
作者：
Mostovyi, Oleksii;Siorpaes, Pietro
设μ，ν是任意可测空间（Ω，F）上的有限正测度，ν= ν a+ ν s是ν关于μ的Lebesgue分解，P是Ω的所有有限分划π <$F的族P，f π（μ）：=∑ A∈ π：μ（A）> 0 1A ν（A）μ（A），π∈ P.我们记得（f π（μ））π∈ P→ d ν ad μ在L1（μ）中，这是（本质上）已知的.这里我们确定（π n）n∈ N <$P使得f π n（μ）→ d ν a d μ μ as as n→∞;在可分F的设定中，一个（相当平凡的）方法已经知道了。为了做到这一切，我们证明了詹森条件不等式中的等式的情况（推广了标准詹森不等式的已知情况），并利用它来确定，给定一个p-一致可积鞅（fi）i∈ I，如何可以确定一个序列（i n）n∈ N <$I，使得（fi n）n在Lp中收敛到某个f，该f闭合整个网（fi）i。我们还给出了一个新的证明（已知）的特征的p-一致可积鞅，不依赖于鞅的收敛定理。
Let μ, ν be positive finite measures on an arbitrary measurable space (Ω, F), ν= ν a+ ν s be the Lebesgue decomposition of ν with respect to μ, P be the family P of all finite partitions π⊆ F of Ω, and f π (μ):=∑ A∈ π: μ (A)> 0 1 A ν (A) μ (A), π∈ P. We recall that (f π (μ)) π∈ P→ d ν a d μ in L 1 (μ), as is (essentially) known. Here we identify (π n) n∈ N⊆ P such that f π n (μ)→ d ν a d μ μ as as n→∞; in the setting of separable F, a (rather trivial) way to do this was already known. To do all this, we characterise the case of equality in Jensen's conditional inequality (generalising the known case of the standard Jensen inequality), and use this to determine how, given a p-uniformly integrable martingale (f i) i∈ I, one can identify a sequence (i n) n∈ N⊆ I such that (f i n) n converges in L p to some f which closes the whole net (f i) i. We also give a new proof of the (already-known) characterisation of p-uniformly integrable martingales, without relying on the martingale as convergence theorem.
影响因子：
1.4
作者：
Beiglboeck, Mathias;Schachermayer, Walter;Veliyev, Bezirgen
通讯作者：
Veliyev, Bezirgen