Question 10.T.19: If the functions f, g : Ω → ¯R are equal almost everywhere a...

Elements of Real Analysis

If the functions $f, g : Ω → \bar{\mathbb{R}}$ are equal almost everywhere and the function f is measurable, then g is also measurable.

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Let

$E = \left\{x ∈ Ω : f (x) ≠ g(x)\right\}.$

Then, for any $α ∈ \mathbb{R},$

$\left\{x : g(x) > α\right\} = (\left\{x : f (x) > α\right\}∩E^{c}) ∪(\left\{x : g(x) > α\right\}∩E).$ (10.19)

Since m(E) = 0, both E and $E^{c}$ are measurable, hence, f being measurable,

$\left\{x : f (x) > α\right\} ∩ E^{c} ∈\mathcal{M}.$

Since m(E) = 0 and {x : g(x) > α} ∩ E ⊆ E, the completeness of m implies that {x : g(x) > α} ∩ E is measurable. By equation (10.19), the set {x : g(x) > α}, and therefore g, is measurable.

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