In these lecture notes by Daniel Bump on Induced Characters I have a question on the proof of Theorem 2.5.1. If $H$ is a subgroup of the finite group $G$ and $(\pi, V)$ a representation of $H$, i.e. a homomorphism $\pi : H \to GL(V)$, then the theorem states a way to get a representation $(\theta^G, V^G)$ on the whole of $G$.
Here if $\theta$ is a function on $H$, the extension to $G$ is done by defining$$ \dot \theta(g) = \left\{ \begin{array}{ll} \theta(g) & \mbox{ if } g \in H,\\ 0 & \mbox{ otherwise} \end{array}\right.$$and then$$ \theta^G(g) = \frac{1}{|H|} \sum_{x\in G} \dot \theta(xgx^{-1}).$$The theorem states:
Theorem: Let $\theta$ be the character of the representation $V$ of $H$. The function $\theta^G$ is the character of a representation of $G$. Its degree is $[G : H]\theta(1)$. If $\chi$ is any character of $G$, and $\chi_H$ is its restriction to $H$, then$$ \langle \theta^G, \chi \rangle_G = \langle \theta, \chi_H \rangle_H.$$In its proof almost everything is clear, how the above formula is verified and so on, but then in the part where he proves that $\theta^G$ is a character I do not understand whats happening. There he states:
It remains to be proved that $\theta^G$ is a character. Let $(\pi_1, V_1), \cdots, (\pi_h, V_h)$ be representatives of the isomorphism classes of the irreducible $G$-modules, and $\chi_1, \cdots, \chi_h$ be their characters. Since the $\chi_i$ are a basis of the complex vector space of class functions, and since $\theta^G$ is a class function, we can write $$ \theta^G = \sum_{i = 1}^h a_i \chi_i, $$ where the $a_i$ are complex numbers, and if we can show that $a_i$ are nonnegative integers, it will follow that $\theta^G$ is a character, namely it will be the character of the representation on the module $$ a_1 V_1 \oplus \cdots \oplus a_h V_h . $$
I do not understand, why if the $a_i$ are nonnegative integers, then $\theta^G$ is a character? And why is it the character of the representation on the module $a_1 V_1 \oplus \cdots \oplus a_h V_h$?