Question

Suppose \([a,b]\subset \large \cup^{\infty}_{k=1} I_k\) .
a) Show that \( \exists n\in \mathbb N\) such that \([a,b]\subset \large\cup^\n_{k=1}I_k\)
b) Let f be the characteristic function of [a,b] and \(f_1,f_2,....,f_k\) be characteristic functions of \(I_1,...,I_k\) (g is called a characteristic function of a set S if g(x) =1 for \(x\in S\) and g(x) =0 otherwise). Prove that for any interval [c,d] that contains \(\large\cup^n_{k=1} I_k\), \(\int_c^d f(x) dx = b-a\) and \(\int_c^d f_k(x) =l(I_k)\)
c) Show that \(f(x)\leq \sum_{k=1}^n f_k(x)\) and use this and the properties of integrals to conclude that \(b-a\leq \sum_{k=1}^n l(I_k)\leq \sum_{k=1}^{\infty} l(I_k)\)
Please, help

Answer 1

@FibonacciChick666

Answer 2

ah... Whats a big union thingy mean again?