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

[seeker101]

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?

 

[LCKurtz]

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.

 

[seeker101]

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?

 

[LCKurtz]

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 A_i are measurable sets. Then A can be rewritten a

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

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