The product of distributions is associative, commutative, and distributive.

I explain that:

In your definition of the product of distributions, you have a summation over all possible products of Dirac delta functions. So, for example, in the case of two distributions A and B, you would have:

A(f) * B(f) = sum over all products of Dirac delta functions of A and B

= sum over all products of Dirac delta functions of B and A

= B(f) * A(f)

So the product of distributions is commutative.

Similarly, for the case of three distributions A, B, and C, you would have:

A(f) * (B(f) * C(f)) = sum over all products of Dirac delta functions of A, B, and C

= sum over all products of Dirac delta functions of (A * B), and C

= (A * B)(f) * C(f)

= A(f) * (B(f) * C(f))

So the product of distributions is associative.

Finally, for the case of four distributions A, B, C, and D, you would have:

A(f) * (B(f) + C(f)) = sum over all products of Dirac delta functions of A and (B + C)

= sum over all products of Dirac delta functions of A, B, and C

= A(f) * B(f) + A(f) * C(f)

= (A * B)(f) + (A * C)(f)

= A(f) * (B(f) + C(f))

So the product of distributions is distributive.