In terms of the Dirac representation of sample mean and the weak convergence of empirical distributions that holds almost surely, we construct a new proof for a strong law of large numbers of Kolmogorov's type with i.i.d. random variables $X_{1}, X_{2}, \dots$ such that $\lim_{c \to \infty}\sup_{n \in \mathbb{N}}n^{-1}\sum_{i=1}^{n}|X_{i}|\cdot \mathbb{I}_{[c,+\infty[}\circ |X_{i}| = 0$ almost surely. That each random variable $X_{i}$ is $L^{1}$ is also a conclusion. Our proof is independent of both Kolmogorov's strong law and its known proof(s), and potentially furnishes a new way to obtain a short proof of Kolmogorov's strong law.