Vtyjkyuk

Let $F: mathbbR to mathbbR$ be any increasing, right-continuous function. Then we define the Lebesgue-Stieltjes measure associated to $F$ to be the unique measure $mu_F$ on $(mathbbR,mathcalB_mathbbR)$ such that $mu_F((a.b])=F(b)-F(a)$ for all $a, b$.

We wish to show that the completion of $mathcalB_mathbbR$ with respect to $mu_F$, denoted by $M_F$, is strictly larger than $mathcalB_mathbbR$.

Since $textcard(mathcalB_mathbbR)=mathfrakc$, it will suffice to exhibit a set $K$ in $M_F$ such that $mu_F(K)=0$ and $textcard(K)=mathfrakc$, for then $textcard(M_F)>mathfrakc$.

We mimic the construction of the Cantor set. For any interval $I$, let us call an open subinterval $J$ of $I$ with $mu_F(J)geqmu_F(I)/2$ and endpoints in the interior of $I$ an "open middle superhalf" of $I$. Observe that, since it is increasing, $F$ has only countably many points of discontinuity. Let $K_1$ be a closed interval with endpoints where $F$ is continuous, and $K_n+1$ be obtained by removing from each of the intervals comprising $K_n$ an open middle superhalf at whose endpoints $F$ is continuous. Then one can check that $K=bigcap_1^inftyK_n$ has the desired properties.