 # I have a question about a finite σ measure on ( <mrow class="MJX-TeXAtom-ORD"> < opepayflarpws 2022-06-25 Answered
I have a question about a finite σ measure on $\left({\mathbb{R}}^{\mathbb{+}},B\left({\mathbb{R}}^{\mathbb{+}}\right)\right)$:

I know that I should use fubini, but unfortunately I don't know where to start. Any help welcome.
You can still ask an expert for help

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it Cahokiavv
Notice that
${\mathbf{1}}_{\left[x,\mathrm{\infty }\right)}\left(y\right)=\left\{\begin{array}{ll}1& y\ge x\\ 0& y
which implies:
${\int }_{{\mathbb{R}}^{+}}x{\mathbf{1}}_{\left[x,\mathrm{\infty }\right)}\left(y\right)dx={\int }_{\left(0,y\right]}xdx=\frac{{y}^{2}}{2}$
So
$\begin{array}{rl}{\int }_{{\mathbb{R}}^{+}}x\mu \left(\left[x,\mathrm{\infty }\right)\right)dx& ={\int }_{{\mathbb{R}}^{+}}{\int }_{{\mathbb{R}}^{+}}x{\mathbf{1}}_{\left[x,\mathrm{\infty }\right)}\left(y\right)\mu \left(dy\right)dx=\\ & \stackrel{\left(1\right)}{=}{\int }_{{\mathbb{R}}^{+}}{\int }_{{\mathbb{R}}^{+}}x{\mathbf{1}}_{\left[x,\mathrm{\infty }\right)}\left(y\right)dx\mu \left(dy\right)=\\ & ={\int }_{{\mathbb{R}}^{+}}\frac{{y}^{2}}{2}\mu \left(dy\right)\end{array}$
where (1) is by Fubini-Tonelli.

We have step-by-step solutions for your answer!