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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.

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    Do you know what a borel space is? – 2017-01-30

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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.

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    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? – 2017-01-30
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    No that is not a nondeterministic function, a non deterministic function would return diffrent values for the same input. – 2017-01-30
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    I see. So it's like using a function with random number generator? – 2017-01-30
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    Yes that is correct. – 2017-01-31
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    Thank you very much for your confirmation. – 2017-02-04