Question: For a random variable X(ω) define the functions X+(ω) = max{X(ω), 0} and X-(ω) = min{X(ω), 0}. Show that X+(ω) and X-(ω) are random variables (i.e., they are measurable functions).

Measurements recorded sequentially in time are often represented graphically as points in the d-dimensional real Euclidean space Rd (d = 1, 2, . . . ,). When the points are sampled from a curve in Rd , they form a path. For example, recordings of trajectories of Brownian particles in R3 reveal that they have continuous paths, however, repeated recordings yield different paths that look completely erratic and random. When tracking charged Brownian particles (e.g., ions in solution) in the presence of an external electrostatic field, the paths remain continuous, erratic, and random; however, they tend to look different from those of uncharged Brownian particles.