Measure Theory

Core idea

Click any term to expand its definition inline.

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Sigma-Algebras

• Set algebra (algebra of sets)
• σ-algebra
• Borel σ-algebra
• Measurable space
• Measurable set
• Measurable rectangle

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Measurable Functions

• Measurable function
• Indicator function
• Simple function

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Measures

Definitions

• Premeasure
• Measure
• Measure space
• Outer measure
• Null set
• Almost everywhere

Constructions

• Carathéodory measurable set
• Carathéodory construction

Properties

• Continuity from above
• Continuity from below

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Lebesgue Measure

• Jordan content
• Lebesgue measure
• Lebesgue criterion for Riemann integrability

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Lebesgue Integration

Basic Definitions

• Almost-everywhere equality
• Lebesgue integral (nonnegative functions)
• Lebesgue integral
• Lebesgue integrable function

L^p Spaces

• Essential supremum
• L^p norm
• L^p space
• L¹ function
• L^∞ function

Convergence

• Convergence almost everywhere
• Convergence in measure
• Convergence in L^p
• Uniform integrability

Product Measures

• Product measure
• Pushforward measure
• Change of variables (pushforward)

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Convergence Theorems

• Monotone Convergence Theorem
• Fatou's Lemma
• Dominated Convergence Theorem
• Tonelli's Theorem
• Fubini's Theorem

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Inequalities

• Jensen's Inequality (integral form)
• Minkowski Inequality (L^p)

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Uncategorized

• Characteristic function (indicator function)
• Lebesgue number lemma refinement lemma
• Set of measure zero in ℝ^k