This question is about the motivation for Weinberg's approach in "The Quantum Theory of Fields" to obtain unitary representations of Lie groups out of its generators.
One is dealing with a Lie group $G$. We have coordinates $\{\theta^a\}$ on a neighborhood of the identity and $T(\theta)$ is the group element with coordinates $\theta$. Group multiplication is encoded in a function $f$ as $$T(\bar{\theta})T(\theta)=T(f(\bar{\theta},\theta)).$$
If $U(T(\theta))$ is a unitary representation on a Hilbert space, the generators of the representation are defined by the expansion $$U(T(\theta))=1+it_a\theta^a+O(\theta^2).$$
The problem then is:
If we know the $t_a$ and how they act on the Hilbert space, how can we find $U(T(\theta))$?
This is dealt with in Weinberg's Appendix 2B:
To prove this theorem, let us recall the method by which we construct the operators corresponding to symmetry transformations. As described in Section 2.2, we introduce a set of real variables $\theta^a$ to parameterize these transformations, in such a way that the transformation satisfy the composition rule (2.2.15): $$T(\bar{\theta})T(\theta)=T(f(\bar{\theta},\theta)).$$ We want to construct operators $U(T(\theta))\equiv U[\theta]$ that satisfy the corresponding condition $$U[\bar{\theta}]U[\theta]=U\left[f(\bar{\theta},\theta)\right].\tag{2.B.1}$$ To do this, we lay down arbitrary 'standard' paths $\Theta_\theta^a(s)$ in group parameter space, running from the origin to each point $\theta$, with $\Theta^a_\theta(0)=0$ and $\Theta_\theta^a(1)=\theta^a$, and define $U_\theta(s)$ along each such path by the differential equation $$\dfrac{d}{ds}U_\theta(s)=it_aU_\theta(s) h^a_{\phantom{a}b}(\Theta_\theta(s))\dfrac{d\Theta^b_\theta(s)}{ds}\tag{2.B.2}$$ with the initial condition $$U_\theta(0)=1,$$ where $$[h^{-1}]^a_{\phantom{a}b}(\theta)=\left[\dfrac{\partial f^a(\bar{\theta},\theta)}{\partial \bar{\theta}^b}\right]_{\bar{\theta}=0}.$$
The basic claim is that if we know the generators $t_a$ of the representation we can find the unitary representation $U[\theta]$ by defining $U_\theta(s)$ through Eq. (2.B.2) and identifying $U[\theta]=U_\theta(1)$.
What is the motivation for Weinberg's approach? What is the motivation to define $U_\theta(s)$ by (2.B.2)? How one would even think about defining this $U_\theta(s)$ through (2.B.2) in order to obtain $U[\theta]$ out of the generators?
