Definition of the Chi-squared distribution (in most of the textbooks) is the following:
If $Z_1,\ldots,Z_k$ are independent standard normal random variables then distribution of the sum of their squares is called the Chi-squared distribution: $\sum_{i=1}^k Z_i^2 \sim \chi^2(k).$
But what about the converse statement?
Let $X \sim \chi^2(k)$ on $(\Omega, \mathcal{F}, P)$. Does it imply that there exists independent standard normal random variables $Z_1, \ldots, Z_k$ on $(\Omega, \mathcal{F}, P)$ such that $X = \sum_{i=1}^k Z_i^2$ ?