I have a similar question as here. Lesbesgue-Stieltjes measure $\mu_F$ with respect to the function $F=\lvert x\rvert\lfloor x\rfloor$. i.e. $\mu_F([a,b))=F(b)-F(a)$. I'm trying to find the Lebesgue-Radon-Nikodym decomposition of $\mu_F$ with respect to the Lebesgue measure $m$. Now, I know thi...

What does the function $f(x,y)$ reduce to? $$f(x,y) = \lim_{n \to \infty} \sum_{i = 0}^{n} (((x \bmod 2^{i-1})-(x \bmod 2^i)) \cdot ((y \bmod 2^{i-1})-(y \bmod 2^i)) \cdot \frac {1}{2^i})$$

I'm going through Folland's Real Analysis, and has come to an impasse trying to come up with a couple examples in the topic of differentiability. On $(\mathbb R, \mathcal B_{\mathbb R}, m)$, for locally integrable $f$, define $F(x)=\int_{0}^x f(t)dt$, $A_rf(x)=\frac{1}{2r}\int_{x-r}^{x+r}f(t)...

I'm going through Folland's Real Analysis, and has come to an impasse trying to come up with a couple examples in the topic of differentiability. On $(\mathbb R, \mathcal B_{\mathbb R}, m)$, for locally integrable $f$, define $F(x)=\int_{0}^x f(t)dt$, $A_rf(x)=\frac{1}{2r}\int_{x-r}^{x+r}f(t)...