Foundations

Scientific relationships · 7 / 7

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

1. Lebesgue integral and integrability → Sets, membership and basic operations
   Uses definition
   Relationship sources · 1

   • Measure Theory, chapter 4: Integration ↗

   Linked statements

   • On a measure space, the integral of a nonnegative measurable f is the supremum of integrals of nonnegative simple functions below f. A real measurable f is integrable when ∫|f|dμ<∞; then ∫f=∫f⁺−∫f⁻.
2. Monotone convergence and Fatou's lemma → Lebesgue integral and integrability
   Uses definition
   Relationship sources · 1

   • Measure Theory, chapter 4: Integration ↗

   Linked statements

   • For increasing nonnegative measurable fₙ→f, ∫fₙdμ→∫f dμ. For arbitrary nonnegative measurable fₙ, ∫lim inf fₙ dμ≤lim inf ∫fₙdμ.
3. Dominated convergence theorem → Lebesgue integral and integrability
   Uses definition
   Relationship sources · 1

   • Measure Theory, chapter 4: Integration ↗

   Linked statements

   • If measurable real fₙ→f pointwise and |fₙ|≤g for one integrable g, then ∫fₙdμ→∫f dμ.
4. Fubini and Tonelli interchange → Lebesgue integral and integrability
   Uses definition
   Relationship sources · 1

   • Measure, Integration & Real Analysis, section 5B: Iterated integrals ↗

   Linked statements

   • On sigma-finite measure spaces, a nonnegative product-measurable function has equal product and iterated integrals, possibly infinite. If its absolute integral is finite, the signed integral is finite and both integration orders give the product integral.
5. Differential entropy and scale → Lebesgue integral and integrability
   Uses definition
   Relationship sources · 1

   • EE/Stats 376A (Winter 2017), Lecture 15: Continuous information measures ↗

   Linked statements

   • For a real density f with defined integral, h(X)=−∫f(x)log₂f(x)dx. It can be negative and changes by log₂|a| under scaling aX, a≠0; therefore it is not the finite bits needed to encode an exact real value.
6. Expectation from a continuous density → Lebesgue integral and integrability
   Uses definition
   Relationship sources · 1

   • STATS 200 (Autumn 2016), Lecture 2: Probability review ↗

   Linked statements

   • For X with density f and measurable g, E[g(X)]=∫R g(x)f(x)dx when ∫|g(x)|f(x)dx<∞. For nonnegative g the expectation may instead be infinite. Finite expectations add linearly without assuming independence.
7. Banach spaces and the role of completeness → Lebesgue integral and integrability
   Related to
   Relationship sources · 2

   • 18.102 (Spring 2021): Banach spaces ↗
   • Measure Theory, chapter 7: L-p spaces ↗

   Linked statements

   • A normed real or complex vector space is Banach when every norm-Cauchy sequence converges in that space. L-p spaces, 1 ≤ p ≤ infinity, use equivalence classes modulo almost-everywhere equality and are complete. C([a,b]) with the supremum norm is complete. In contrast, continuous ramps on [0,1] can converge in L1 to a step with no continuous representative, so the L1 norm does not make that continuous-function space complete.

   L-p spaces depend on integrability and almost-everywhere equivalence.