These are the problems, that I can´t do, from my book. They are interesting. Please help me.
i) Let $f\colon[a,b] \to \mathbb{R}$ be Lipschitz (in particular, it is $\mathcal{C}^1$). If $ X \subset [a,b] $ has null measure, then $f(X)$ has null measure.
ii) Show that if $ f,g:[a,b] \to \mathbb{R}$ are Riemann-integrable, and the set $ X = \left\{x\ \left|\ f(x) \ne g( x)\right\}\right. $ has measure zero, then $\int_a^b f( x )\,dx = \int_a^b g( x )\,dx$ where the integrals are the Riemann integrals.
Here's one thought: separate the integration zone into $X$ and $[a,b]-X$, but in Riemann integrals, this may not make sense. We integrate only on intervals.
I'm very confused with this problem, and I don´t know how to do it formally