The characteristic Function of SVC, $ \chi$ is not Riemann integrable as proved here

But it can be shown that $ \chi $ is Lebesgue integrable. $ $ \int_0^1 \chi d\mu=\int_{SVC} \chi d\mu+\int_{[0,1]\setminus {SVC}} \chi d\mu=\int_{SVC} d\mu=\mu(SVC)=\frac{1}{2}.$ $ Question 1:- Is this correct?

Now to prove $ \chi$ is HK- integrable,

Let $ \epsilon>0$ be any real number.

Consider the $ \gamma$ -fine Tagged partition $ P=\{\{ c_k,[t_{k-1},t_k] \} : 1 \le k \le n\}$ with $ $ \delta(x)=\left\{ \begin{array}{ll} \frac{1+\epsilon}{2^{j+1}} & x=c_k=e_j \in \text{SVC} \ 1 & x\in [0,1] \setminus \text{SVC} \end{array}\right. $ $ Where $ e_j:$ the end point of the $ j$ th removed interval from $ [0,1]$ to form the SVC. $ $ \therefore |S(f,P)-1/2|= |\sum_{c_k \in SVC} f(c_k)\Delta t_k + \sum_{c_k \notin SVC} f(c_k)\Delta t_k -1/2|$ $

$ $ =|\sum_{c_k \in SVC} f(c_k)\Delta t_k -1/2|\le|\sum_{j=1}^{\infty} \frac{1+\epsilon}{2^{j}} -1/2|= \epsilon$ $ Hence $ $ HK\int_0^1 \chi =1/2$ $ Question 2:- Is this correct?