Mean of empty set? How to take care of it in proofs?

[Merudo]

How do I efficiently take care of the case where I'm taking the mean of a set if the set can be empty?

\(\displaystyle \sum_{i \in A} \frac{b_i}{|A|} \)

The previous equation is not well defined when the set is empty. So I tried this


\(\displaystyle I[ |A| > 0 ]\sum_{i \in A} \frac{b_i}{|A|} \)

But this is still undefined! Is there anything I can do to transform the undefined mean to an actual number?

I guess I could use piecewise analysis, such as

\(\displaystyle f(A) = \begin{cases}
\sum_{i \in A} \frac{b_i}{|A|} & \text{if } |A| > 0 \\
0 & \text{if } |A| = 0
\end{cases} \)

However this rapidly becomes cumbersome - if an equation has k potentially undefined means, it means I need to take care of 2^k different cases.

 

[HallsofIvy]

Actually, "f(A)= 0 for |A|= 0" is wrong. if |A|= 0, f(A) is undefined.

 

[Merudo]

Actually, "f(A)= 0 for |A|= 0" is wrong. if |A|= 0, f(A) is undefined.

How come the following function is undefined?

\(\displaystyle f(A) = \begin{cases}
\sum_{i \in A} \frac{b_i}{|A|} & \text{if } |A| > 0 \\
0 & \text{if } |A| = 0
\end{cases} \)

 

[HallsofIvy]

I didn't say the function, f, is undefined, I said that the value, f(A), when |A|= 0, is undefined. The empty set does NOT have a mean while you are saying the mean is 0.

 

[Merudo]

I didn't say the function, f, is undefined, I said that the value, f(A), when |A|= 0, is undefined. The empty set does NOT have a mean while you are saying the mean is 0.

I know that, I'm trying to go around that fact so that I have a different mean that gives 0 when the set is empty.