By Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski

As a part of the clinical job in reference to the seventieth birthday of the Adam Mickiewicz college in Poznan, a world convention on algebraic topology was once held. within the ensuing complaints quantity, the emphasis is on monstrous survey papers, a few awarded on the convention, a few written as a consequence.

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**Example text**

21 Let (Ω, Σ, μ) be a measure space and let {un }n 1 ⊆ L1 (Ω) be a sequence such that un −→ u in μ-measure or for μ-almost all ω ∈ Ω for some u ∈ L1 (Ω). 22 Let (Ω, Σ, μ) be a measure space and let {un }n 1 ⊆ Lp (Ω) (with 1 < p < +∞) be an Lp -bounded sequence such that un (ω) −→ u(ω) for μ-almost all ω ∈ Ω. 23 Let (Ω, Σ, μ) be a ﬁnite measure space and let {un }n 1 ⊆ L1 (Ω) be a uniformly integrable sequence such that un (ω) −→ u(ω) for μ-almost all ω ∈ Ω. Show that un −→ u in L1 (Ω). 24 Suppose that (Ω, Σ, μ) is a ﬁnite measure space, u ∈ L1 (Ω), {un }n 1 ⊆ w is a sequence such that un −→ u in μ-measure and un −→ u in L1 (Ω).

We deﬁne W01,p (Ω) = Cc∞ (Ω) · 1,p . When p = 2, we write H01 (Ω) = W01,2 (Ω). Evidently W01,p (Ω) ⊆ W 1,p (Ω) and it is a Banach space (Hilbert if p = 2) equipped with the W 1,p (Ω)-norm. 121, we know that W01,p (RN ) = W 1,p (RN ). If Ω ⊆ RN , then in general the inclusion W01,p (Ω) ⊆ W 1,p (Ω) is strict. However, if RN \ Ω is “thin” and p < N , then W01,p (Ω) = W 1,p (Ω) (for example when Ω = RN \ {0}). 38 Chapter 1. 128 If Ω ⊆ RN is a set with a Lipschitz boundary ∂Ω and u ∈ W 1,p (Ω) ∩ C(Ω) with 1 p < +∞, then the following statements are equivalent: (a) u ∈ W01,p (Ω).

Show that u is Bochner integrable and u(ω) X dμ lim inf un (ω) n→+∞ Ω X dμ Ω (a version of the Fatou lemma for the Bochner integral). 63 Let (Ω, Σ, μ) be a ﬁnite measure space, X and Y two Banach spaces, D ⊆ X a linear subspace, L : D −→ Y a closed linear operator (recall that closed means that Gr L ⊆ X × Y is closed) and u : Ω −→ X a Bochner integrable function such that L ◦ u : Ω −→ Y is Bochner integrable too. Show that for all A ∈ Σ we have u dμ L A = L(u) dμ. 2. 64 Let (Ω, Σ, μ) be a ﬁnite measure space, X a Banach space and u, y : Ω −→ X two Bochner integrable functionals such that u(ω) dμ = y(ω) dμ A ∀ A ∈ Σ.