# $\sigma(\boldsymbol{\mathfrak C}) / \boldsymbol{\mathfrak B}_\mathbb R$ measurable function on $\boldsymbol{\mathfrak C}=\{[n,n+1):n\in\mathbb Z\}.$

In $$\S 1.4$$ of Real Analysis: Theory of Measure and Integration by J. Yeh, the author provided an example:

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

Now, it's easy to see that functions of such type are $$\sigma(\boldsymbol{\mathfrak C}) / \boldsymbol{\mathfrak B}_\mathbb R$$ measurable. But I am not able to formalize how it is the other way. Any hint would be appreciated.

• Measurable functions are constant on atoms. Apr 23, 2023 at 12:20

If $$f$$ is $$\sigma(\boldsymbol{\mathfrak C})$$-measurable then $$f^{-1} (\{f(n)\})$$ is a union of the (disjoint) intervals $$[k,k+1)$$ and it contains $$n$$. Hence it must contain $$[n,n+1)$$. Can you finish?
• Appreciate your quick comment and the answer. Now to consolidate my thoughts: $f$ is constant a.e. on each atom $[n, n+1)$ and if $[k, k+1) \in f^{-1} (\{f(n)\})\implies k = n$ and subsequently we get the required step function that is right continuous at each integer. Please correct me if I am wrong. Apr 23, 2023 at 16:16