At this paper, page $128$, in Theorem $3.1$, there are two parts which I don't understand how the authors got them:
$(1)$ In the first sentence of the proof, why the set $$K=\{ \sum_{n=1}^{\infty}{t_nx_n}:0 \leq t_n \leq 1 \}$$ is a norm compact subset of $X$?
$(2)$ Why $\beta(\phi_n)=x_n$?
For (1), let $(y_n)$ be a dense sequence in $S_X$ (the unit sphere of $X$), define $(x_n)$ by $x_n := 2^{-n}y_n$, then the span of $(x_n)$ is dense in $X$ and $K_0 := \{x_n \mid n \in \mathbf N\} \cup \{0\}$, is compact, as the sequence $x_n$ converges to $0$. Now, $K_1 := \operatorname{\overline{conv}}K_0$ is still compact, as $K$ is a closed subset of $K_1$, $K$ is also compact.
For (2), note that $\beta \colon \mathcal F(X) \to X$ is linear and continuous, hence \begin{align*} \beta(\phi_n) &= \beta\left(\int_{H_n} \delta\bigl(x_n + S_n(t)\bigr) - \delta\bigl(S_n(t)\bigr)\, d\lambda_n(t)\right)\\ &= \int_{H_n} \beta\Bigl(\delta\bigl(x_n + S_n(t)\bigr) - \delta\bigl(S_n(t)\bigr)\Bigr)\, d\lambda_n(t)\\ &= \int_{H_n} \beta\delta\bigl(x_n + S_n(t)\bigr) - \beta\delta\bigl(S_n(t)\bigr)\, d\lambda_n(t)\\ &= \int_{H_n} \bigl(x_n + S_n(t) - S_n(t)\bigr)\,d\lambda_n(t)\\ &= \int_{H_n} 1 \, d\lambda_n(t)\, \cdot x_n\\ &= x_n \end{align*}