New MT Article Ideas

Here is a list of new article ideas for the Measure Theory Wiki.

Analysis Fundamentals

• Metric space vs. Pseudometric space

Measurable Sets

• Vitali's Theorem and non-existence of a measure on $2^{\mathbb {R} }$
• Borel $\sigma$-algebra and ${\mathcal {B}}_{\mathbb {R} }$
• Product $\sigma$ - algebra
• Borel measureable sets vs. Lebesgue measurable sets

Measures

• Caratheodory's Theorem
• Isomorphism of Measure Spaces
• Lebesgue-Stieljes Measures
• Characterization of Finite Borel Measures
• Approximation by Open/Compact Sets

Measurable Functions

• Simple functions
• Measurable functions

Integration of Nonnegative Functions

• Monotone Convergence Theorem
• Fatou's Lemma
• Key counterexamples

Integration of Real-Valued Functions

• Dominated Convergence Theorem
• L1 as a normed vector space

Modes of Convergence

• Pointwise a.e. convergence
• L1 convergence
• Convergence in measure
• Egorov's theorem