Each colored bar is a disjoint measurable set. The total measure (blue bar) equals the sum of individual measures — this is countable additivity. Add more sets to see the sum grow.
ν(A) = ∫_A f(x) dμ(x), f(x) = e^{-(x-1)²/0.5}
The red curve is the Radon–Nikodym derivative dν/dμ. The shaded area under the curve over [a,b] gives ν([a,b]). The base measure μ is Lebesgue measure (uniform). Adjust the scale to see how the density reshapes the measure.
λ*([0.2, 0.7]) = inf Σ(bᵢ−aᵢ) = 0.50
The green region is the target set. Blue intervals form a countable cover. The outer measure is the infimum of total lengths of all such covers. Add intervals to cover the target; the total length (shown) is an upper bound on λ*.