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Is the following reasonment correct?

There is a sort of duality between non-decreasing functions and Borel outer measures.

In particular, given a non-decreasing function $f:\mathbb{R}\to\mathbb{R}$, the associated Lebesgue-Stieltjes outer measure $\lambda_f$ is a Radon outer measure.

Viceversa, given $\mu$ finite Borel outer measure measure, the so called distribution function associated to $\mu$, $f(x):=\mu((-\infty,x])$, is non-decreasing and right-continuous.

My questions are:

  1. If we do not require $\mu$ to be Borel, what will happen to the distribution $f$?
  2. If we do not require $\mu$ to be finite, what will happen to the distribution $f$?
  3. If we require $\mu$ to be also Radon, what will happen to the distribution $f$?
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