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Is this :

sage: integrate(g(x),(x,-oo,oo)).full_simplify().canonicalize_radical()         
-e^(1/2*a^2*s^2 - a*m)

what you want ?

BTW :

sage: assume (a, m, x, "real")                                                  
sage: assume(s>0)                                                               
sage: assumptions()                                                             
[a is real, m is real, x is real, s > 0]
sage: integrate(g(x),(x,-oo,oo)).full_simplify()                                
-e^(1/2*a^2*s^2 - a*m)
sage: integrate(g(x),(x,-oo,oo)).simplify()                                     
-e^(1/2*a^2*s^2 - a*m)

In integration, assumptions may be capital...