# 实分析 Real Analysis MATH20101

conclusion holds for $k=1$. Let us proceed by induction on $k$. If $C$ has no interior, then by Theorem $6.2 .6$ it is included in a hyperplane $H=f^{-1}{c}$ for some non-zero linear form $f$ and $c \in \mathbb{R}$. Then
$$H={x: f(x) \geq c} \cap{x: f(x) \leq c},$$
an intersection of two half-spaces. Also,
$$H=u+V:={u+v: v \in V}$$

Let $V$ be a real vector space and $C$ a convex set in $V$. A real-valued function $f$ on $C$ is called convex iff
$$f(\lambda x+(1-\lambda) y) \leq \lambda f(x)+(1-\lambda) f(y)$$

$$t \leq 3 \sum_{i=1}^{q} \lambda\left(J_{i}\right) \leq 6 j \sum_{i=1}^{q} \mu\left(J_{i}\right) \leq 6 j \mu(V) \leq 6 j \varepsilon .$$
Thus $\lambda\left(P_{j}\right) \leq 6 j \varepsilon$. Letting $\varepsilon \downarrow 0$ gives $\lambda\left(P_{j}\right)=0, j=1,2, \ldots$, and letting $j \rightarrow \infty$ proves the lemma.
Now to prove Theorem 7.2.1, for each rational $r$ let
$$g_{r}:=\max (g-r, 0), \quad f_{r}(x):=\int_{a}^{x} g_{r}(t) d t .$$