English
The integral over an interval in AddCircle T equals the integral over the corresponding interval in the lift by the quotient map.
Русский
Интеграл по интервалу в AddCircle(T) равен интегралу по соответствующему интервалу в воздухе посредством отображения-фрактирования.
LaTeX
$$$\\int_{a}^{a+T} f(x)\\,dx = \\int_{\\mathrm{AddCircle}(T)} f\\,db$$$
Lean4
/-- The integral of an almost-everywhere strongly measurable function over `AddCircle T` is equal
to the integral over an interval (t, t + T] in `ℝ` of its lift to `ℝ`. -/
protected theorem intervalIntegral_preimage (t : ℝ) (f : AddCircle T → E) :
∫ a in t..t + T, f a = ∫ b : AddCircle T, f b :=
by
rw [integral_of_le, AddCircle.integral_preimage T t f]
linarith [hT.out]