Why is the sum of two simple functions also a simple function?

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Discussion Overview

The discussion revolves around the properties of simple functions, specifically focusing on whether the sum of two simple functions remains a simple function. Participants explore both intuitive and formal aspects of this concept, including considerations of mutual exclusivity in the definition of simple functions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants express an intuitive understanding that the sum of two simple functions is also simple, seeking a formal proof to support this claim.
  • Others argue that the definition of a simple function as a linear combination of indicator functions of measurable sets inherently supports the idea that their sum is also a simple function.
  • A later reply introduces a modified definition requiring the events to be mutually exclusive and questions how this affects the proof that the sum remains simple.
  • Another participant suggests that the problem can be reformulated in terms of measurable sets and proposes an inductive approach to prove the equivalence of the two forms.

Areas of Agreement / Disagreement

Participants generally agree on the intuitive notion that the sum of two simple functions is simple, but the introduction of mutual exclusivity leads to differing views on how to formally prove this, indicating unresolved aspects of the discussion.

Contextual Notes

The discussion highlights the dependence on definitions and the implications of mutual exclusivity, which may affect the validity of claims regarding the sum of simple functions.

seeker101
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seeker101 said:
I can intuitively see why the sum of two http://en.wikipedia.org/wiki/Simple_function#Definition" is also simple. But can someone point me to a formal proof?

What is there to prove? The definition says a simple function is a linear combination of indicator functions of measurable sets. A sum of two of them is just a longer linear combination of indicator functions of measurable sets.
 
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seeker101 said:
Sorry. I left something out.

Suppose now the definition of a http://en.wikipedia.org/wiki/Simple_function#Definition" also requires the events A_k to be mutually exclusive.

How can we now show that the sum of two simple functions will also be simple?

It looks equivalent to me to the following statement:

Suppose:

A = \cup_{i=1}^n A_i

where the Ai are measurable sets. Then A can be rewritten a

A = \cup_{i=1}^n B_i

where the Bi are disjoint measurable sets. And this looks trivial to prove by induction.
 
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