PROBLEM SET 3, 18.155DUE MIDNIGHT FRIDAY 30 SEPTEMBER, 2011(1) Consider the space L1loc(Rn) of locally Lebesgue integrable func-tions, defined byf ∈ L1loc(Rn) ⇐⇒ fR(x) =(f(x) if |x| < R ∀ R > 00 if |x| ≥ R∈ L1(Rn),L1loc(Rn) = L1loc(Rn)/a.e.Show that the kfNkL1, N ∈ N, form a countable set of semi-norms with respect to which L1loc(Rn) is complete, i.e. it is com-plete with respect to the metricd(f, g) =XN2−NkfN− gNkL11 + kfN− gNkL1.(2) Prove the Monotonicity Lemma. If fj∈ L1(Rn) are real-valued,fj(x) is monotonic increasing for each x andRfjis boundedthen there exists f ∈ L1(Rn) such that fj(x) → f(x) a.e. andlimj→∞Rfj→Rf.(3) We say that U ⊂ Rnis Lebesgue measurable if the characteristicfunction χU(x) = 1 if x ∈ U, χU(x) = 0 otherwise is an elementof L1loc(Rn). It is of finite measure if χU∈ L1(Rn) and themeasure (volume) of U is the integral. Show that the Lebesguemeasurable sets form a σ-algebra – the collection of measurablesets is closed under complements (in Rn) and countable unions(and hence countable intersections).(4) Give an example of a continuous function u ∈ C0(Rn) which isnot polynomially bounded yet is such that(1) C∞c(Rn) 3 φ 7−→Zu(x)φ(x)extends by continuity to an element of S0(Rn). Hint: The sim-plest example I know is the derivative of a function like exp(iex).(5) Show that the Fourier transform defines a continuous linear mapF : L1(Rn) −→ C00(Rn)= {u : Rn−→ C; u is continuous and limR→∞sup|x|≥R|u(x)| = 0}.12 PROBLEM SET 3, 18.155 DUE MIDNIGHT FRIDAY 30 SEPTEMBER, 2011Hint: Check it for a dense subspace, such as S(Rn), and thenget an estimate which proves
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