1991 Nov

  1. a. Let \(f_n\) be a sequence of continuous, real valued functions on \([0, 1]\) which converges uniformly to \(f\). Prove that \(\lim_{n → ∞} f_n (x_n) = f(1/2)\) for every sequence \(\{x_n\}\) that converges to \(1/2\).
    b. Must the conclusion still hold if the convergence is only point-wise? Explain.

  2. Let \(f : ℝ → ℝ\) be differentiable and assume there is no \(x ∈ ℝ\) such that \(f(x) = f'(x) = 0\). Show that \(S = \{x : 0 ≤ x ≤ 1, f(x) = 0\}\) is finite.

  3. If \((X, Σ, μ)\) is a measure space and if \(f\) is \(μ\)-integrable, show that for every \(ϵ > 0\) there exists \(E ∈ Σ\) such that \(μ(E) < ∞\) and \(∫_{X - E} |f| \, dμ < ϵ\).

  4. If \((X, Σ, μ)\) is a measure space, \(f\) is a non-negative measurable function, and \(ν(E) = ∫_E f \, dμ\), show that \(ν\) is a measure.

  5. Suppose \(f\) is a bounded, real valued function on \([0, 1]\). Show that \(f\) is Lebesgue measurable if and only if \(\sup ∫ ψ \, dm = \inf ∫ ϕ \, dm\), where \(m\) is Lebesgue measure and \(ϕ\) and \(ψ\) range over all simple functions for which \(ψ ≤ f ≤ ϕ\).

  6. If \(f\) is Lebesgue integrable on \([0, 1]\) and \(ϵ > 0\), show that there exists \(δ > 0\) such that for all measurable sets \(E ⊂ [0, 1]\) with \(m(E) < \delta\), we have \(| ∫_E f\, dm | < ϵ\).

  7. Suppose \(f\) is a bounded, real valued, measurable function on \([0, 1]\) such that \(∫ x^n f\, dm = 0\) for \(n = 0, 1, 2, …\), where \(m\) is Lebesgue measure. Show that \(f(x) = 0\) a.e.

  8. If \(μ\) and \(ν\) are finite measures on the measurable space \((X, Σ)\), show that there is a nonnegative measurable function \(f\) on \(X\) such that for all \(E\) in \(Σ\), \(∫_E (1 - f)\, dμ = ∫_E f\, dν\).

  9. If \(f\) and \(g\) are integrable functions on \((X, S, μ)\) and \((Y, T , ν)\), respectively, and \(F(x, y) = f (x) g(y)\), show that \(F\) is integrable on \(X × Y\) and \(∫ F \, d(μ × ν) = ∫ f \, dμ ∫ g\, dν\).