Departments of Mathematics
Fall 2015
Montana State University
Prof. Kevin Wildrick
Measure Theory
Review Set 3
1. Repeat Reviews 1 and 2.
2. Make a list of all homework problems assigned but not completed. Do them.
3. Define
• Lp (X, Σ, µ) for 1 ≤ p ≤ ∞. (note: we did not discuss L∞ (X, Σ, µ) in class, but it’s
definition was assigned reading)
• convergence in Lp (X, Σ, µ)
• convergence pointwise almost-everywhere
• convergence in measure
• nearly uniform convergence
• a charge
• positive, negative, and null sets for a charge
• a Hahn decomposition of a charge
• the positive and negative parts of a charge, and the absolute value of a charge
• absolute continuity of one measure with respect to another
• singularity of one measure with respect to another
• a bounded linear function between normed spaces
• a positive bounded linear function on Lp (X, Σ, µ)
4. State and prove
•
•
•
•
•
•
the Lebesgue dominated convergence theorem
Hölder’s inequality
Minkowski’s inequality
Egoroff’s theorem
the Hahn decomposition theorem
the Radon Nikodym theorem
5. Let {fn }n∈N and f be measurable functions on (X, Σ, µ) such that fn → f pointwise
µ-almost everywhere. Show that for 1 ≤ p < ∞,
||fn − f ||p → 0 if and only if||fn ||p → ||f ||p .
6. Formulate and prove a version of Hölder’s inequality for the product of three functions.
7. Suppose that µ is a finite measure on ((0, ∞), B) such that
• µ m,
• µ(aB) = µ(B) for each a ∈ (0, ∞) and Borel set B in B, where
aB = {ab : b ∈ B}
• the Radon-Nikodym derivative
dµ
dm
is a continuous function.
Show that there is a constant c ≥ 0 such that
dµ
dm (x)
= xc .