equivalent intervals of measurability for measurable functions

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Theorem

Let E⊂R be measurable and let f:E→[−∞,+∞]. Then the following are equivalent for all α∈R,

  1. f−1((α,∞])∈M
  2. f−1([α,∞])∈M
  3. f−1([−∞,α))∈M
  4. f−1([−∞,α])∈M
Proof

Not hard!

(1)⟹(2)

Suppose (1) holds. Then for all α∈R

[α,∞]=⋂n(α−1n,∞]⟹f−1([α,∞])=⋂nf−1((α−1n,∞])

RHS is a countable intersection of measurable sets and is thus measurable.

(2)⟹(1)

Suppose (2) holds. Then for all α∈R,

(α,∞]=⋃n[α+1n,∞]⟹f−1(α,∞]=⋃nf−1([α+1n,∞])

And this is a countable union of measurable sets and is thus Lebesgue measurable

ie (1)⟺(2)

(3)⟺(4) uses the same argument.

(2)⟺(3)

Note that [−∞,α)=([α,∞])c. Thus by taking the preimages and noting that complements of measurable sets are measurable yields the desired result.

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