Analysis? Set theory?

For a subset of real numbers $S$, let $\mathbf{1}_S : \mathbb{R} \to \{0, 1\}$ be the indicator function of $S$ , defined as $\mathbf{1}_S(x) = 1$ if $x \in S$ and $\mathbf{1}_S(x) = 0$ otherwise.

Prove or disprove: for every real function $f : \mathbb{R} \to \mathbb{R}$ , there exists subsets of real numbers $A_1, A_2, A_3, \ldots$ and real numbers $c_1, c_2, c_3, \ldots$ such that

$\displaystyle\large{ f(x) = \sum_{n=1}^\infty c_n \mathbf{1}_{A_n}(x) }$

for all real $x$ .

Clarification: When I posted this problem, I didn't know the answer. Now, I found the answer, but I find it interesting (like most set theory stuff), so I'll let you to figure it out.

#Algebra

Easy Math Editor

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Comments

Given a real number, we can express its binary representation as $\dots b_2 b_1 b_0.b_{-1} b_{-2} \dots$ , where each digit $b_i$ is 0 or 1.

For an integer $i$ , let $S_i$ be the set of real numbers $x$ such that the $i$ th digit in the binary representation of $f(x)$ is equal to 1. Then $f(x) = \sum_{i \in \mathbb{Z}} 2^i \mathbf{1}_{S_i}(x).$

Jon Haussmann - 4 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$