# Equivalent definitions of a content zero set

According to Spivak (before thrm 3.5) ,

A subset of $$\ \mathbb{R^n}$$ has ($$n$$ - dimensional) content zero if for every $$\epsilon > 0$$ there is a $$finite$$ cover $$\ \{U_1 , U_2 , \cdots,U_m \}$$ by closed rectangles such that $$\sum_{i=1}^m v(U_i) < \epsilon$$ .

Also according to Spivak (after thrm 3.9 ),

A bounded set C whose boundary has measure 0 is called Jordan Measurable . The integral $$\displaystyle \int_C 1$$ is called the ($$n$$ - dimensional ) content of C

Are these two definitions equivalent ie ; is the following true :

A set is of content zero iff content of the set is zero ?

Here is a proof of the forward implication.

Since $$C$$ is of content zero, it is bounded and is contained inside a bounded rectangle $$Q$$. The content of $$C$$ is the Riemann integral of $$f \equiv 1$$ over $$C$$, defined as

$$\int_Cf= \int_Q g,$$

where $$g(x) = 1$$ for $$x \in C$$ and $$g(x) = 0$$ for $$x \in Q \setminus C$$.

Note that $$g$$ is Riemann integrable, since it is continuous everywhere except possibly at points of $$C$$ and the content of $$C$$ is zero.

Now take any partition $$P$$ of $$Q$$. Any subrectangle $$R$$ of $$P$$ has non-zero content (by definition of a partition). Hence $$R$$ is not a subset of $$C$$ and must contain at least one point where $$g(x) = 0$$. This implies that $$\inf_R g(x) \leqslant 0 \leqslant \sup_R g(x)$$. Forming upper and lower Riemann sums we have for any partition $$P$$,

$$L(P,g) \leqslant 0 \leqslant U(P,g).$$

Since $$g$$ is integrable we have

$$\int_Q g = \sup_P L(P,g) \leqslant 0 \leqslant \inf_P U(P,g) = \int_Q g,$$

And this implies that $$\int_Q g = 0$$, i.e., the content of C is zero.

For the reverse implication, if the content of $$C$$ is zero then $$\int_Q g =0$$ for any enclosing rectangle $$Q$$. Hence, for any $$\epsilon >0$$ there exists a partition $$P$$ such that $$0 \leqslant U(P,g) < \epsilon$$. Considering only the subrectangles that intersect $$C$$, we can show that these subrectangles form a finite covering of $$C$$ with total volume less than $$\epsilon$$. See if you can fill in the details of this last step in the proof.