 Solution

For a maximum of finitely many nonnegative quantities, its $p$th power is at most the sum of their $p$th powers. Hence the increment assumption givesBy the triangle inequality in the Lp norm, the partial sums defining $K_{\alpha }$ have norms bounded by a geometric series. Monotone convergence of their $p$th powers givesIn particular $K_{\alpha }$ is finite almost surely, so the preceding pathwise estimate has a common probability-one domain for all dyadic pairs.