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A function u : X → R is said to be measurable with respect to A if u−1 (E) ∈ A for every Borel set E ⊂ R. , uk ∈ R, uk χk (x), u(x) = χk (x) = 1 for x ∈ Ek , χk (x) = 1 otherwise, k and {Ek } ⊂ A is a ﬁnite collection of pairwise disjoint measurable sets covering X. Notice that for any measurable function u : X → R there is an increasing sequence of simple functions u(n) such that u(n) (x) ≤ u(n+1) (x), lim u(n) (x) = u(x) in X. 2) A simple function u ≥ 0 is integrable if uk μ(Ek ) < ∞. u dμ := X k Here, we adopt the standard convention 0 · ∞ = 0 for μ(Ek ) = ∞.

For all 0 < s < 1 and 1 < r < ∞, we denote by W0s,r (Ω) the interpolation space [W00,r (Ω), W01,r (Ω)]s,r endowed with one of the equivalent norms deﬁned by the real interpolation method. In other words, W0s,r (Ω) is obtained by real interpolation of the subspaces W01,r (Ω) = W01,r (Ω) ⊂ W 1,r (Ω) and W00,r (Ω) = Lr (Ω). 11, the space W01,r (Ω) is dense in W0s,r (Ω) for 0 < s < 1 < r < ∞. It follows that C0∞ (Ω) is dense in W0s,r (Ω) and hence W0s,r (Ω) is the completion of C0∞ (Ω) in the W0s,r -norm.

24, μ can be represented as a weak measurable family of Radon measures μx . e. x ∈ Ω. To this end choose an arbitrary nonnegative function ψ ∈ C0 (Ω), and select η ∈ C0 (R) with 0 ≤ η ≤ 1, and η(x) = 1 for |x| ≤ N . We have (1 − ε) ψ(x) dx ≤ Ω ψ(x) dx − ψ(x) Ω Ω ψ(x) = R Ω R (1 − η(x/N )) dμnx dx η(x/N ) dμnx dx ≤ ψ(x) dx. Ω Letting n → ∞ we obtain (1 − ε) ψ(x) dx ≤ Ω ψ(x) R Ω η(x/N ) dμx dx ≤ ψ(x) dx. Ω Letting N → ∞ and next ε → 0 we arrive at ψ(x) Ω R ψ(x) dx for all nonnegative ψ ∈ C0 (Ω). e.