The steps below show that the family (epsilon_{lambda}(t):=e^{-lambda t}, lambda, t>0), is determining for (([0, infty), mathscr{B}[0, infty))). (i) Weierstra' approximation theorem shows that the

 The steps below show that the family \(\epsilon_{\lambda}(t):=e^{-\lambda t}, \lambda, t>0\), is determining for \(([0, \infty), \mathscr{B}[0, \infty))\).

(i) Weierstraß' approximation theorem shows that the polynomials on \([0,1]\) are uniformly dense in \(C[0,1]\), see Theorem 28.6 .

(ii) Define for \(u \in C_{c}[0, \infty)\) the function \(u \circ(-\log ):(0,1] ightarrow \mathbb{R}\) and approximate it with a sequence of polynomials \(p_{n}, n \in \mathbb{N}\).

(iii) Show that \(\int \epsilon_{\lambda} d \mu=\int \epsilon_{\lambda} d u\) implies \(\int p_{n}\left(e^{-t}ight) \mu(d t)=\int p_{n}\left(e^{-t}ight) u(d t)\); thus, \(\int u d \mu=\int u d u\).

Data from theorem 28.6

Theorem 28.6 (Weierstra) Polynomials are dense in C[0, 1] w.r.t. uniform convergence. Proof (S. N. Bernstein) Take a sequence (i)EN of independent measur- able functions on ([0, 1], B[0, 1], dx) which are Bernoulli (p. 1 - p)-distributed, 0