How can I prove that $BV([a,b])$ is not separable. Here, $BV([a,b])$ means all the function defined on $[a,b]$ with finite total variation.
2026-03-26 09:25:28.1774517128
Why $BV[a,b]$ is not separable?
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The total varation distance between $I_{(x, \frac {1+x} 2)}$ and $I_{(y, \frac {1+y} 2)}$ is $1$ whenever $x \neq y$ ($x,y \in (0,1)$). Hence the space is not separable.