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

1. Apr 11, 2010

### seeker101

Last edited by a moderator: Apr 25, 2017
2. Apr 11, 2010

### LCKurtz

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.

Last edited by a moderator: Apr 25, 2017
3. Apr 11, 2010

### 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?

Last edited by a moderator: Apr 25, 2017
4. Apr 11, 2010

### LCKurtz

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.

Last edited by a moderator: Apr 25, 2017