Departments of Mathematics
Fall 2015
Montana State University
Prof. Kevin Wildrick
Measure Theory
Problem Set 11
Due Friday, November 20
th ,
11:00 am.
The symbol (?) indicates that this problem must be solved and turned in. Other problems
should be solved but need not be turned in.
1. Let λ be charge on a set X equipped with a σ-algebra Σ. Show that λ+ and λ− are
independent of the choice of the Hahn decomposition used to define them.
2. (?) Exercises 8A and 8B in Bartle.
3. (?) Let (X, Σ, µ) be a measure space, and let f ∈ L1 (X, Σ, µ). Define µf : Σ → R by
Z
µf (E) :=
f dµ.
E
• Show that µf is a charge.
• Do the sets
P = {x ∈ X : f (x) ≥ 0} and N = {x ∈ X : f (x) < 0}
define a Hahn decomposition for µf ?
• Show that for any E ∈ Σ,
µ+
f (E)
Z
f
=
E
+
dµ, and
µ−
f (E)
Z
=
E
f − dµ.