May I know the intuitive understanding of a deterministic, non-negative, and Borel-measurable function?

Especially, I am not sure what the 'deterministic' and 'Borel-measurable' functions are.

Could you please help me understand this in the most easiest way?

Thank you.

  • $\begingroup$ Do you know what a borel space is? $\endgroup$ – Q the Platypus Jan 30 '17 at 0:36

A Borel (sub)set is a set that can be created from open sets by countable union, countable intersection and relative complement. A Borel-measurable function is one where every preimage of an interval is a borel set.

In other words if $ f : \mathbb{R}^k \to \mathbb{R} $ is a borel measurable function then for every interval $\Delta$, $f^{-1}[\Delta]$ is a borel subset of $\mathbb{R}^k$.

Deterministic functions are functions that always give the same value for the same attributes in other words functions that are nonrandom.

  • $\begingroup$ Thank you so much. Then correct me if I am wrong. Can I call the function nondeterministic if it reruns 1 if x is greater than 5 and 0 otherwise? Can I assume that a function that can give identical values for a random number like this will be nondeterministic? $\endgroup$ – Eric Jan 30 '17 at 11:21
  • $\begingroup$ No that is not a nondeterministic function, a non deterministic function would return diffrent values for the same input. $\endgroup$ – Q the Platypus Jan 30 '17 at 12:34
  • $\begingroup$ I see. So it's like using a function with random number generator? $\endgroup$ – Eric Jan 30 '17 at 12:35
  • $\begingroup$ Yes that is correct. $\endgroup$ – Q the Platypus Jan 31 '17 at 3:53
  • $\begingroup$ Thank you very much for your confirmation. $\endgroup$ – Eric Feb 4 '17 at 19:26

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