From equality of integrals to equality of functions #
This file provides various statements of the general form "if two functions have the same integral on all sets, then they are equal almost everywhere". The different lemmas use various hypotheses on the class of functions, on the target space or on the possible finiteness of the measure.
Main statements #
All results listed below apply to two functions
f, g, together with two main hypotheses,
gare integrable on all measurable sets with finite measure,
- for all measurable sets
swith finite measure,
∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ. The conclusion is then
f =ᵐ[μ] g. The main lemmas are:
ae_eq_of_forall_set_integral_eq_of_sigma_finite: case of a sigma-finite measure.
ae_fin_strongly_measurable.ae_eq_of_forall_set_integral_eq: for functions which are
Lp.ae_eq_of_forall_set_integral_eq: for elements of
0 < p < ∞.
integrable.ae_eq_of_forall_set_integral_eq: for integrable functions.
For each of these results, we also provide a lemma about the equality of one function and 0. For
We also register the corresponding lemma for integrals of
ℝ≥0∞-valued functions, in
Generally useful lemmas which are not related to integrals:
ae_eq_zero_of_forall_inner: if for all constants
λ x, inner c (f x) =ᵐ[μ] 0then
f =ᵐ[μ] 0.
ae_eq_zero_of_forall_dual: if for all constants
cin the dual space,
λ x, c (f x) =ᵐ[μ] 0then
f =ᵐ[μ] 0.
Don't use this lemma. Use