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Prove that the function is monotonic and not decreasing


New member
Jun 16, 2016
Hello, I don't know to solve this exercise:

Let $\mathcal{B}_\mathbb{R}$ the $\sigma-algebra$ Borel in $\mathbb{R}$ and let $\mu : \mathcal{B}_\mathbb{R} \rightarrow{} \mathbb{R}_{+}$ a finite measure. For each $x \in \mathbb{R}$ define

$$f_{\mu} := \mu((- \infty,x]) $$

Prove that:

a) $f_{\mu}$ is a monotonic non-decreasing function
b) $\mu((a,b]) = f_{\mu}(b)- f_{\mu}(a)$ for all $a,b \in \mathbb{R}$

The definition ($\sigma-algebra$ borel , is this:

Definition $\sigma-algebra$ Borel : Let $(X,\tau)$ a topological space. we define the borelian tribe associated with $(X, \tau)$ as the algebra generated by T, that is

$$\mathcal{B}(X)= \mathcal{B}(X,\tau)= \sigma (\tau)$$

I need a hint.