Upcrossing number of an interval

For a real sequence $x=(x_n{)}_{n\ge 0}$, reals $a<b$, and $N\ge 0$, define $U_N[a,b](x)=\max \{k:\mathrm{\exists }0\le s_1<t_1<\cdots <s_k<t_k\le N,x_{s_i}\le a,x_{t_i}\ge b\},$ where the empty tuple makes $0$ admissible, and put $U_{\mathrm{\infty }}[a,b](x)={\sup }_{N\ge 0}{U}_N[a,b](x)\mathrm{.}$ For a real process $X$, these definitions are applied pathwise.

For fixed $N$ there are finitely many candidate tuples. For each $k$, the event $\{U_N\ge k\}$ is a finite union of finite intersections of events $\{X_s\le a\}$ and $\{X_t\ge b\}$, hence is measurable. Thus the integer-valued $U_N$ is measurable. Since $U_N\uparrow U_{\mathrm{\infty }}$, measurability of the countable pointwise supremum follows from Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable. No optimizing tuple is selected, so the definition is choice-free.