Problem Set 15

1. Medium
Let $$Z_i \subseteq \mathbb R^N$$, $$i=1,\ldots,n$$ be a collection of zero Jordan measure sets. Show that the union $$\bigcup Z_i$$ also has zero Jordan measure.

2. Medium
If $$S = \left\{x_1,\ldots,x_n\right\}$$ is a finite set consisting of precisely $$n$$-elements, show that $$S$$ has zero Jordan measure.

3. Hard
Let $$f:[a,b] \to \mathbb R$$ be a bounded, integrable function.
1. Show that the graph of $$f$$, $$\Gamma(f) := \left\{(x,f(x)): x \in [a,b]\right\}\subseteq \mathbb R^2$$ has zero Jordan measure.
2. If $$f$$ is non-negative, show that $$S = \left\{(x,y): a \leq x \leq b, 0 \leq y \leq f(x)\right\}$$ is measurable, and $$m(S) = \int_a^b f(x) dx$$.

4. Hard
Show that if $$f: \mathbb R \to \mathbb R^2$$ is a $$C^1$$ function, then for any interval $$I \subseteq \mathbb R$$, $$f(I)$$ has zero Jordan measure.

5. Hard
Let $$f:[a,b] \to \mathbb R$$ be a Riemann integrable function. If $$g:[a,b] \to \mathbb R$$ is another function and $$S = \left\{x: f(x) \neq g(x)\right\}$$ contains exactly $$n$$-points, show that $$g$$ is also Riemann integrable. [Note: You must prove this from scratch. If you wish to invoke a corollary or result from class, you must first prove it.]

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