Definition 6.1.1 (Median): We say that $c$ is a median of a random variable $X$ if $\mathbb{P}(X≥m)≥ 1/2$ and $\mathbb{P}(X≤m)≥ 1/2$. (The simplest way this can happen is if the CDF of $X$ hits $1=2$ exactly at $c$, but we know that some CDFs have jumps.)
Why the definition put the $<$ and $>$ with equal before $1/2$? and what is the relation between the median and the sentence between the brackets