 Time-zero correction for a half-point dyadic spike martingale

On $(0,1]$ with Lebesgue probability and the usual dyadic filtration, each displayed support is a dyadic atom. Its next split has one child of doubled height and one of zero height, preserving the conditional average. Hence the sequence is a martingale from time one. To extend it to the trivial time-zero sigma-algebra one must set $X_0=1$: applying the same displayed formula at zero instead gives $1_{(0,1/2]}$, which is not adapted and has the wrong expected value. The repaired martingale tends to zero almost surely while its expected values remain one, so it is not uniformly integrable.