Question

Let   be the distribution function defined by   

Let  $\mu _{F}$ be the Lebesgue-Stieltjes measure asociated to .

Determine the measurements  of the fpllowing sets:

$\left \{ 2 \right \}$

$[-\frac{1}{2},3)$

Answer 1

Solution Given function is ro if x.4-1 F(1 = htxe if IERO : Tex² if osaa2 19 if x22 following Nao we have determine the sets measurement S. (V {2} le { {2}} = F(2) F(2) = 9-2 +22) 39- (2+4) - 9.-6 (2} [3, 3) = ((-1,3)] = F (5)+F(64))

- 1- (1) = 1-() 아을 - 905 ( (-1, 0] (1,2) ㅋ 4 ((-1,]) = F (0) F (1) = 2 0 22. LA (1,2)) = f (2) - F ) = (2+2) ~ (2+3) = (+y-2M = 63 | = 3 . . ( (-10]UCt,2)) = + 3 = 15 |

(4) [otju (1,2] → (10, 1)) = f))-FC0") = (2+)-1 → u (12]) = F (21)-F (1) = 9-3 . u ([0, b) U(1,2]) = +6 = 5+24

53 (x: +22 > When, so, > JU1호의 (E, JJ) = F (J) F(-) ⇒ 살 | ((, )) )) - F (c) - F( - -- 4 675

St.9+5.0 = st:9+ = (loc. {]n (10-))