Measure theory

• Is There a Compact Set That Is Not Jordan Measurable?
  Yes. The Smith–Volterra–Cantor set is compact, has positive measure, and has empty interior, so its boundary is not null and it is not Jordan measurable.
• How Does Expected Value Change with the Probability Measure?
  Expected value depends on the probability measure: integrate the same random variable under each measure, or use a density to convert between them.
• How Is the Convolution of an L1 and an L2 Function Defined?
  For f in L1 and g in L2, define their convolution almost everywhere by an integral; Young's inequality guarantees it belongs to L2.
• Why Are Constant Functions Measurable?
  A constant function is measurable because every measurable-set preimage is either the whole domain or the empty set, both in any sigma-algebra.
• How Are Borel Measures Related to Functions of Bounded Variation?
  A right-continuous function of bounded variation defines a finite signed Borel measure through its interval increments; measures recover such functions.
• How to Approximate Simple Functions by Step Functions Almost Uniformly
  On a finite interval, approximate each measurable level set by finite unions of intervals; summable errors make the resulting step functions converge almost uniformly.
• A Finitely Additive Measure Is Countably Additive Exactly When It Is Continuous from Below
  A nonnegative finitely additive set function on a sigma-algebra is countably additive if and only if its values converge along every increasing sequence of sets.
• When Is a Random Variable Almost Surely Constant?
  A real random variable independent of itself must be almost surely constant; see the proof, a Bernoulli example, and other useful criteria.
• How to Prove the Main Properties of Conditional Expectation
  Prove linearity, positivity, the tower property, and other conditional expectation identities from its defining integral property.