Exam 5 (36–42)¶

1998 Jun¶

Instructions Attempt at least five of the following problems. Partial credit will be given for significant progress toward solutions. You may use standard results which have been established in courses, but indicate what result you are using when you use it.

Problem 36

The limit supremum and limit infimum of a sequence of sets \(\{E_n\}\) are defined to be

\[\varlimsup_n E_n = ⋂_n \left(⋃_{k≥ n} E_k\right) \quad \text{ and } \quad \varliminf_n E_n = ⋃_n \left(⋂_{k≥ n} E_k\right).\]

Prove that \(\varliminf_n E_n ⊆ \varlimsup_n E_n\).
Given an example of a sequence of sets for which \(\varliminf_n E_n ≠ \varlimsup_n E_n\).
If the sets \(E_n\) are measurable subsets of a measure space \((X, 𝔐, μ)\), make and prove the strongest statement possible about the relationship between \(m \left(\varliminf_n E_n\right)\) and \(\varliminf_n m(E_n)\).

Problem 37

Prove: If \(f\) and \(g\) are continuous real-valued functions on \([0,1]\), and \(f(x) = g(x)\) for almost all \(x\), then \(f(x) = g(x)\) for all \(x\).

Problem 38

Let \(ℕ\) denote the set of natural numbers and \(\{p_n\}\) a summable sequence of nonnegative real numbers. For \(E⊆ N\), define \(μ(E) = ∑_{n∈ E}p_n\).

Show that \(μ\) is a measure on the subsets of \(ℕ\).
Give a formula for \(∫_ℕ f \, dμ\) where \(f\) is a bounded real-valued function on \(ℕ\).

Problem 39

Let \((X, 𝔐, μ)\) be a measure space and assume \(\{f_n\}_{n=1}^∞\) is a sequence of nonnegative measurable functions such that \(f_n → f_0\) a.e. and \(∫_X f_n \, dμ → 0\). Show that \(∫_X f_0 \, dμ = 0\).

Problem 40