# Proof of ∪∞k=1Ek=E\cup_{k=1}^{\infty}E_k =E for measurable sets EE and EkE_k

Let , fn:E→[0,∞]f_n:E\to [0,\infty] be a monotone increasing sequence of measurable functions converges pointwise to ff ; where EE is a measurable set. Choose a simple function ϕ:E→[0,∞)\phi:E\to [0,\infty) such that ϕ≤f\phi \le f on EE and choose a real number a(0 Nn > N, a\phi(x) \leq f_n(x) \leq f(x)a\phi(x) \leq f_n(x) \leq f(x).
– Hans
2 days ago

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