English
If r ≠ ∞, then μ.withDensity (r • f) = r • μ.withDensity f.
Русский
Если r не бесконечно, то μ.withDensity (r • f) = r • μ.withDensity f.
LaTeX
$$$r \neq \infty \Rightarrow \mu.withDensity (r \cdot f) = r \cdot \mu.withDensity f$$$
Lean4
theorem withDensity_smul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) : μ.withDensity (r • f) = r • μ.withDensity f :=
by
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs, smul_eq_mul, ←
lintegral_const_mul' r f hr]
simp only [Pi.smul_apply, smul_eq_mul]