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.


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