Why does the Dirac comb function multiply the sum of Dirac delta functions by T?

Consider what happens as T goes to zero. What happens if you integrate the product with and without the leading factor of T?

Still don't quite understand why we need to multiply the sampling function with T

Same question... I cannot figure out why we need a T here...

Why does the Dirac comb function multiply the sum of Dirac delta functions by T?

Consider what happens as T goes to zero. What happens if you integrate the product with and without the leading factor of T?

Still don't quite understand why we need to multiply the sampling function with T

Same question... I cannot figure out why we need a T here...