# Characterization measurability of a function in a particular sigma-algebra

I am trying to prove the next:

Let $$\boldsymbol{C} = \{[n, n + 1) : n\in\mathbb{Z}\}.$$ The $$\sigma$$-algebra generated by $$\boldsymbol{C}$$, $$\sigma(\boldsymbol{C})$$, is the collection of all countable unions of members of $$\boldsymbol{C}$$. An extended real-valued function defined on $$\mathbb{R}$$ is $$\sigma(\boldsymbol{C})$$-measurable if and only if it is a right-continuous step function with jump discontinuity occurring at integers in $$\mathbb{R}$$ only.

I have troubles with the part "if and only if": If the function is right-continuous step function with jump discontinuity is measurable because is the limit of linear combination of step function over subsets of the class $$\boldsymbol{C}.$$ For the other implication it seems "intuitively" true but I cannot see why; disjointness of the class could be the fact of $$f$$ has jump discontinuity at integers and be a step function but I am stuck in this.

Any kind of help is thanked in advanced.

Edit: The definition of measurability that I am following is:

Let $$(X,\mathcal{A})$$ be an arbitrary measurable space and let $$D\in \mathcal{A}.$$ An extended real-valued function f defined on $$D$$ is said to be $$\mathcal{A}$$-measurable on $$D$$ if it satisfies the condition that $$\{x\in D: f(x)\leq \alpha\}\in\mathcal{A}$$, that is, $$f^{-1}([-\infty,\alpha])\in\mathcal{A}$$, for every $$\alpha\in\mathbb{R}.$$

• Following a comment from Robert W. on my answer, can you precise on which $\sigma$-algebras are equipped the domain and codomain of your functions? – mathcounterexamples.net Sep 16 at 16:42
• I've put the definition of measurability that I'm using. – Suiz96 Sep 16 at 17:04

## 1 Answer

The result seems to be wrong. Consider the map $$f$$ defined as $$f(x)=\begin{cases} 1/4 & x \le 0\\ 1/4 + x/2 & 0 < x \le 1\\ 3/4 & 1 < x \end{cases}$$

$$f$$ is not a step function. However for $$X \in \sigma(\boldsymbol{C})$$, $$f^{-1}(X) = \mathbb R$$ if $$[0,1) \subseteq X$$ and $$f^{-1}(X) = \emptyset$$ otherwise. Proving that $$f$$ is $$\sigma(\boldsymbol{C})$$ measurable.

• Many thanks! That was a very nice counterexample. By the way, Your website is awesome!! – Suiz96 Sep 16 at 16:27
• Thanks for the compliment. Really appreciated! – mathcounterexamples.net Sep 16 at 16:29
• It seems that the question is about $\sigma(\boldsymbol{C})/\mathcal{B}(\mathbb{R})$ measurability. – Robert W. Sep 16 at 16:40