I'm reading the book "Markov Chains : Gibbs Fields, Monte Carlo Simulation, and Queues" written by Pierre Brémaud In p.333 the author define transition semigroup as this :
Let $(P_{t})_{t \in \mathbb{R}_{+}} \in ([0, 1]^{\mathbb{N}\times \mathbb{N}})^{\mathbb{R}_{+}}$ such that
- $P_{0}(i, j) = \delta_{i, j}$ for every $i, j \in \mathbb{N}$ where $\delta_{i, j}$ is the kronecker symbol.
- $P_{s+t} = P_{s}P_{t}$ for every $s, t \in \mathbb{N}$ (where $P_{s}P_{t}$ is the matrix multiplication)
- For every $t \in \mathbb{R}_{+}, i \in \mathbb{N}$, $\sum_{j \in \mathbb{N}}P_{t}(i, j) = 1$.
Then it's said that if (right-continuous at 0) : $\lim_{t \to 0^{+}} P_{t}(i, j) = \delta_{i, j}$ for every $i, j \in \mathbb{N}$, we have (continuous on $\mathbb{R}_{+}$) : $\lim_{h \to 0} P_{t+h}(i, j) = P_{t}(i, j)$ for every $t \in \mathbb{R}_{+}, i,j \in \mathbb{N}$.
How can we prove this statement ? The right-continuity for every $t \in \mathbb{R}_{+}$ is clear for me, but I'm blocked by left-continuity.