Measurable Function - Non-measurable Functions

Non-measurable Functions

Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to find non-measurable functions.

So long as there are non-measurable sets in a measure space, there are non-measurable functions from that space. If (X, Σ) is some measurable space and A ⊂ X is a non-measurable set, i.e. if A ∉ Σ, then the indicator function 1_A: (X, Σ) → R is non-measurable (where R is equipped with the Borel algebra as usual), since the preimage of the measurable set {1} is the non-measurable set A. Here 1_A is given by

$\mathbf{1}_A(x) = \begin{cases} 1 & \text{ if } x \in A \\ 0 & \text{ otherwise} \end{cases}$

Any non-constant function can be made non-measurable by equipping the domain and range with appropriate σ-algebras. If f: X → R is an arbitrary non-constant, real-valued function, then f is non-measurable if X is equipped with the indiscrete algebra Σ = {0, X}, since the preimage of any point in the range is some proper, nonempty subset of X, and therefore does not lie in Σ.

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