Convergence of Nonmonotone Sequence of Sub-σ-Fields and Convergence of Associated SubspacesLp(Bn) (p∈[1,+∞])

Abstract

Kudō [Ann. Probab. 2(1) (1974), 76-83] introduced a notion of convergence of sub-σ-fields in terms of L 1 -convergence of conditional expectation. We study relationships between convergence of sub-σ-fields and convergence of associated subspaces L p n . In particular, we show that the Slice convergence is not adapted to the L 1 -case in contrast to the Mosco convergence. Notions of upper and lower limits are also studied. This enables us to obtain measurability results regarding those limits. We also give examples and counterexamples in order to exhibit characteristic behaviours of the sequences n n , L p n n , and Ɛ n n .

A New Version of the Multivalued Fatou Lemma

“…(i) =⇒ (ii): Using intrinsic properties of the conditional expectation operator, we obtain (ii) when E = R (see [19,Theorem 2.3.1] for a complete proof). (i) =⇒ (ii): Using intrinsic properties of the conditional expectation operator, we obtain (ii) when E = R (see [19,Theorem 2.3.1] for a complete proof).…”

“…When p ≥ 1, B n → B ∞ if and only if (E Bn (f )) n converges to E B∞ (f ) for every f ∈ L p R (F) (with L p R (F) endowed with the strong topology) (see Kudō [17] when p = 1 ; Piccinini [19] when p ∈ [1, +∞] and for other topologies). When p ≥ 1, B n → B ∞ if and only if (E Bn (f )) n converges to E B∞ (f ) for every f ∈ L p R (F) (with L p R (F) endowed with the strong topology) (see Kudō [17] when p = 1 ; Piccinini [19] when p ∈ [1, +∞] and for other topologies).…”

Compact convergence of σ-fields and relaxed conditional expectation

“…See Tsukuda [34], Artstein [6], Piccinini [28]. The novelty in the present paper is the use of spaces of measure valued maps.…”

“…As noted in Proposition 6.5 the sn p -convergence coincides with the L p -convergence on ordinary σ -fields. Convergence in the L p -norm of the conditional expectation and the connection to convergence of σ -fields were examined in the literature, notably by Dang-Ngoc [18], Fetter [21], Alonso and Brambila-Paz [5], Piccinini [28]. Proof.…”

“…A useful property is the continuity of the conditional expectation operator with respect to the convergence of the σ -fields. A non exhaustive list of papers that address the issue of distance among σ -fields, the continuity of conditional expectation and of other operations is: Allen [1], [2], Allen and Van Zandt [3], Alonso [4], Alonso and Brambila-Paz [5], Artstein [6], Boylan [15], Cotter [16], [17], Dang-Ncog [18], Fetter [21], Kudo [24], Neveu [26], [27], Piccinini [28], Rogge [30], Stincombe [31], [32], Tsukada [34].…”