\displaystyle\int_{\mathbb{R}}g_a^+ \,d\lambda=\int_{1}^\infty g_a^+(x) \,dx=\lim_{n\to \infty}\int_{1}^{n} g_a^+(x) \,dx=\lim_{n\to \infty} \sum_{k=1}^{n-1} \int_{k}^{k+1} a^k \cdot 1_{[k,k+1/2\rangle}(x)\,dx