If $f(x) = \sum_{j=1}^n c_j 1_{A_j}(x)$ ie a simple function. Why is $$\prod_{j=1}^n \exp(t[\exp(i(u,c_j))-1]\mu(A_j)) =\exp[t\int_A[\exp(i(u,f(x)))-1]\mu(dx)].$$ The notation $(u,c)$ is dot product, $1_{A}(x)$ is the indicator function and $i=\sqrt{-1}$.
Asked
Active
Viewed 214 times
3
-
1There is no $x$ in your $f(x)$. Now, what exactly are all those variables and subscripts supposed to mean? The question as it stands is awfully vague. – J. M. ain't a mathematician Oct 12 '10 at 08:58
-
1The $j$ on the right side does not make sense because on the left side you multiply over $j$. – Rasmus Oct 12 '10 at 15:05
-
apologies, I have edited the question. – Vaolter Oct 12 '10 at 18:18
1 Answers
0
Because $e^a \cdot e^b = e^{a+b}$.
The only thing $t(e^{i\langle u,c_j\rangle}-1)$ does (for the purpose of evaluating the integral) is just to make the manipulations a bit more ugly.
kahen
- 15,760