Question 10.T.16: For a function f : Ω → ¯R, the following statements are equi...
For a function f : Ω → \bar{\mathbb{R}}, the following statements are equivalent:
(i) f is measurable.
(ii) For all α ∈ \mathbb{R}, \left\{x ∈ Ω : f (x) > α \right\} is measurable.
(iii) For all α ∈ \mathbb{R}, \left\{x ∈ Ω : f (x) ≥ α \right\} is measurable.
(iv) For all α ∈ \mathbb{R}, \left\{x ∈ Ω : f (x) < α \right\} is measurable.
(v) For all α ∈ \mathbb{R}, \left\{x ∈ Ω : f (x) ≤ α \right\} is measurable.
(i)⇒(ii): We have
\left\{x ∈ Ω : f (x) > α \right\} = f^{−1}((α, ∞)) ∪ f^{−1}(\left\{∞\right\}).
Since f is measurable and (α, ∞) is a Borel set, f^{−1}(\left\{∞\right\}) and f^{−1}((α, ∞)) are both measurable, hence so is {x ∈ Ω : f(x) > α}.
(ii)⇒(i): By Theorem 10.3, the intervals \left\{(α, ∞) : α ∈ \mathbb{R}\right\} generate \mathcal{B}.
Furthermore, the sets
f^{−1}(\left\{∞\right\}) = \overset{\infty}{\underset{n=1}{\cap}} \left\{x ∈ Ω : f (x) > n\right\},
f^{−1}(\left\{−∞\right\}) = \overset{\infty}{\underset{n=1}{\cap}} \left\{x ∈ Ω : f (x) ≤ −n \right\} = \overset{\infty}{\underset{n=1}{\cap}} \left\{x ∈ Ω : f (x) > −n \right\}^{c}
are both measurable. Since
f^{−1}((α, ∞)) = \left\{x ∈ Ω : f (x) > α\right\} \setminus \left\{x ∈ Ω : f (x) = ∞\right\}
is also measurable, we conclude that f is measurable by Lemma 10.5.
The equivalence of (i) and (iii) is established by choosing \mathcal{D} to be composed of all sets of the form [α, ∞), and using Lemma 10.5. To prove the equivalence of (ii) and (v) and that of (iii) and (iv), we resort to the fact that a σ-algebra is closed under complementation.