By putting the product inside the first pair of square brackets you've made the formula look a little bit confusing - since the only factors that depend on j and m_n are outside that parenthesized factor, but I'll give it a try...
The inner integral over x is just the error-function:
(erf(y/2^(1/2)) + 1)/2
This should give you a modified function, something like this:
fun = @(y) ((erf((2^(1/2)*y)/2) + 1))/2.*exp(-1/2*(y+(u_j-u_m)/s_n).^2);
Which for your desired values of u_j, u_m and s_n will integrate just fine from -inf to inf.
HTH
