Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / M / 3 / d / Solution

Equality forces $\mathbb{E}X<\infty$ because its proposed value is at most $a$. The random variable $Y=X-a{1}_{\{X\ge a\}}$ is nonnegative and has expectation zero, so $Y=0$ almost surely. On $\{X<a\}$ it equals the strictly positive variable $X$, making that event null. On $\{X\ge a\}$ it equals $X-a$, forcing $X=a$ there. HenceThis is the equality in Markov's inequality under strict positivity. The reasoning even works without the discreteness assumption.