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A nonnegative measurable function with finite integral is finite almost everywhere
Statement
If is measurable and , then for almost every .
Facts & Assumptions
Given: A nonnegative measurable function with finite integral.
The nonnegative integral is monotone and homogeneous (Monotonicity and nonnegative homogeneity of the nonnegative integral).
The nonnegative integral of the simple function equals its simple integral (The nonnegative integral agrees with the simple integral on simple functions).
Proof
Let , which is measurable. For every positive integer , . By [L1] and [L2], .
If , the inequalities in step 1.1 fail for sufficiently large ; if , they fail already for . Thus , so the exceptional set where is infinite is null. Equivalently, almost everywhere.
Depends on
Used by
- Pointwise restriction is not defined on Lp equivalence classes Counterexample
- Finite and countable planar sets have zero logarithmic capacity Example
- Direct integrals of measurable Hilbert fields are Hilbert spaces Theorem
- Dominated convergence Theorem
- Every sigma-finite signed measure admits a Lebesgue decomposition relative to a sigma-finite positive measure Theorem
- Hardy–Littlewood–Sobolev fractional integration inequality Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John K. Hunter, Measure Theory Notes, Proposition 4.14 (standard reference, not scraped)
- Gerald B. Folland, Real Analysis, 2nd ed., Proposition 2.20 (standard reference, not scraped)