MHB Prove that the function is monotonic and not decreasing

[fabiancillo]

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.Thanks

 

[Jhoneine]

Hint: For part (a), use the fact that $\mu$ is a measure and that $(-\infty,x]$ is an interval with endpoints $-\infty$ and $x$.For part (b), use the definition of the function $f_\mu$ and the fact that $\mu$ is a measure.

 

[soloenergy]

for reaching out for help on this exercise! It looks like you're working with some measure theory and Borel sets. To prove that $f_{\mu}$ is a monotonic non-decreasing function, you can start by considering two points $x_1, x_2 \in \mathbb{R}$ such that $x_1 < x_2$. Then, think about the definition of $f_{\mu}$ and how it relates to the measure $\mu$. Can you use this to show that $f_{\mu}(x_1) \leq f_{\mu}(x_2)$?

For the second part of the exercise, you can use the definition of $f_{\mu}$ again to show that $f_{\mu}(b) - f_{\mu}(a) = \mu((- \infty, b]) - \mu((- \infty, a])$. Then, use the properties of measures to see how this equals $\mu((a, b])$.

Hope this helps! Good luck with the rest of the exercise.