Unexpected result in a simple Integral

Questions
1

I entered

Integral[Exp[I x t - a (u-x) ]  , {x,-Infinity,  u}]  // Simplify

%% (Where I stands for the imaginary unit i)
And got the correct answer but with a weird condition:

E^(I t u)/(a+I t) if Re(a)>Im(t)\[And]Re(a (u+1000345))>=1000345 Im(t)\[And]Re(a (u+1000503))>=1000503 Im(t)

Looks as if infinity equals the number 1000345 ?

The purpose here was to find the characteristic function of a Laplace distribution.

2

I suppose it is a condition for convergence of the integral. A simple example of divergence:

With[{t = 0, u = 0, a = 0},
 Integrate[Exp[I x t - a (u - x)], {x, -Infinity, u}]]

We may find more examples this way:

sol = FindInstance[Not[Re[a] > Im[t] &&
    Re[a (1000076 + u)] >= 1000076 Im[t] &&
    Re[a (1000303 + u)] >= 1000303 Im[t]],
  {t, u, a}, 2]
Exp[I x t - a (u - x)] /. sol[[1]] // ComplexExpand