Ytukyg

Let $Omega := [0,1] times [0,1]$. Consider on $sigma$-algebra Borel sets with Lebesgue measure .Let $X(w)$ describe the distance (in maximum metric) between the point and the nearest corner of the square. Is $X(w)$ a random variable? If so, find probability distribution and CDF.
I need help,beacuse it is quite difficult.

Hint

Let $omega =(omega _1,omega _2)in Omega $. Then,$$X(omega )=infBig{sup{omega _1,omega _2}, sup{omega _1,1-omega _2 }, sup{1-omega _1,1-omega _2}, sup{1-omega _2,omega_2}Big}.$$

Added

Set
begin{align*}
X_1(omega )&=sup{omega _1,omega _2}\
X_2(omega )&=sup{omega _1,1-omega _2}\
X_3(omega )&=sup{1-omega _1,omega _2}\
X_4(omega )&=sup{1-omega _1,1-omega _2}.
end{align*}

$$mathbb P{Xleq x}=mathbb P{Xboldsymbol 1_{[0,1/2]^2}leq x}+mathbb P{Xboldsymbol 1_{[0,1/2]times [1/2,1]}leq x}+mathbb P{Xboldsymbol 1_{[1/2,1]times [0,1/2]}leq x}+mathbb P{Xboldsymbol 1_{[1/2,1]^2}leq x}.$$
Now $$mathbb P{Xboldsymbol 1_{[0,1/2]^2}leq x}=mathbb P{omega in [0,1/2]^2mid X_1leq x}=mathbb P{omegain [0,1/2]^2mid omega _1leq x, omega _2leq x }=mathbb P{omega _1in [0,1/2]mid omega _1leq x}mathbb P{omega _2in [0,1/2]mid omega _2leq x},$$
the last inequality come from independence of $omega _1mapsto omega _1$ and $omega _2mapsto omega _2$. Notice that that r.v. follow uniform law on $[0,1/2]$.

The rest goes the same.