Problem Set 15

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.

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.

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 \(m(S) = \int_a^b f(x) dx\).

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.

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.]