# Documentation

Mathlib.MeasureTheory.Measure.Lebesgue.Integral

# Properties of integration with respect to the Lebesgue measure #

theorem volume_regionBetween_eq_integral' {α : Type u_1} [] {μ : } {f : α} {g : α} {s : Set α} (f_int : ) (g_int : ) (hs : ) (hfg : ) :
↑(MeasureTheory.Measure.prod μ MeasureTheory.volume) () = ENNReal.ofReal (∫ (y : α) in s, (g - f) yμ)
theorem volume_regionBetween_eq_integral {α : Type u_1} [] {μ : } {f : α} {g : α} {s : Set α} (f_int : ) (g_int : ) (hs : ) (hfg : ∀ (x : α), x sf x g x) :
↑(MeasureTheory.Measure.prod μ MeasureTheory.volume) () = ENNReal.ofReal (∫ (y : α) in s, (g - f) yμ)

If two functions are integrable on a measurable set, and one function is less than or equal to the other on that set, then the volume of the region between the two functions can be represented as an integral.

theorem Real.integrable_of_summable_norm_Icc {E : Type u_1} {f : } (hf : Summable fun n => ) :

If the sequence with n-th term the the sup norm of λ x, f (x + n) on the interval Icc 0 1, for n ∈ ℤ, is summable, then f is integrable on ℝ.

### Substituting -x for x#

These lemmas are stated in terms of either Iic or Ioi (neglecting Iio and Ici) to match mathlib's conventions for integrals over finite intervals (see intervalIntegral). For the case of finite integrals, see intervalIntegral.integral_comp_neg.

theorem integral_comp_neg_Iic {E : Type u_1} [] (c : ) (f : E) :
∫ (x : ) in , f (-x) = ∫ (x : ) in Set.Ioi (-c), f x
theorem integral_comp_neg_Ioi {E : Type u_1} [] (c : ) (f : E) :
∫ (x : ) in , f (-x) = ∫ (x : ) in Set.Iic (-c), f x