Download Math 524 Homework: Metric Spaces and Measures and more Assignments Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
Math 524
Homework due 11/08/
Reading
Section 1, Chapter 2 in Folland.
Reminder:
Midterm on Wednesday November 8th.
Problems from Folland:
Chapter 2, Section 1: problems 1, 2 and 9.
Problem 1. Let X = {f : [0, 1] → R; f piecewise continuous}. We say that f is piecewise continuous on [0, 1] if f has only finitely many discontinuities in [0, 1]. For f , g ∈ X we say that f ∼ g is f − g ≡ 0 on [0, 1], except for finitely many points. This defines an equivalence relation on X. Let Y = {[f ] : f ∈ X} where [f ] denotes the equivalence class of f ∈ X. For [f ] , [g] ∈ Y define
d([f ], [g]) =
0
|f − g| dx.
Show that d defines a metric on Y. Show that the closed ball of radius 1 and center [0], i.e.
{[f ] ∈ Y : d([f ], [0]) ≤ 1 }
is not compact.
Problem 2. Le X be a metric space, let B be the Borel σ-algebra over X. A measure μ is said to be outer regular if for every Borel set E
μE = inf {μO : E ⊂ O, O open}.
It is said to be inner regular if
μE = sup {μK : K ⊂ E, K compact}.
The measure μ is Borel regular if it is both inner and outer regular.
Assume that μ is a σ-finite Borel regular measure on a metric space X.
2.1 Show that for every Borel set E and every > 0 there exists an open set O such that E ⊂ O and μ(O\E) <
2.2 Show that for every Borel set E, such that μ(E) < ∞, and every > 0 there exists a compact set K such that K ⊂ E and
μ(E\K) <
∗ Problem: Caratheodory’s criterion Let μ be an outer measure on Rn. If
μ(A ∪ B) = μ(A) + μ(B), ∀ A, B ⊂ Rn^ with d(A, B) > 0 ,
then μ is a Borel measure (i.e. Borel sets are μ-measurable). Here
d(A, B) = inf{|a − b| : a ∈ A, b ∈ B}.
