Equality of sum of multiplications with the same factor

Question

Suppose I know that the following equality holds: $$ \sum_{x} A(x)C(x) = \sum_{x} B(x)C(x) $$

Can I conclude that: $$ \sum_{x} A(x) = \sum_{x} B(x) $$

Does this hold for any function C(x) or a given function C(x)? — failure, May 03 '22 at 16:10
@failure I am considering the case where $C(x) = \log \left( \frac{B(x)}{D(x)} \right)$, for arbitrary probability distributions A, B, C and D. — Gilberto Cunha, May 03 '22 at 16:13

score 0 · Accepted Answer · answered May 03 '22 at 16:43

From

$$\sum_{x} A(x)C(x) = \sum_{x} B(x)C(x)$$

you can conclude that

$$\sum_{x} [A(x)-B(x)]C(x) = 0\tag{1}$$

But from this you cannot conclude that

$$\sum_{x} [A(x)-B(x)]= 0\tag{2}$$

That is, you cannot conclude that

$$\sum_{x} A(x)=\sum_{x}B(x)$$

If (1) implied (2) then, logically it would also imply

$$\sum_{x} C(x)=0$$

But clearly, that is not the case.

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