The integers between 1000 and 9999, inclusive, are all four-digit positive integers of the form $abcd$. We want each of $a, b, c$, and $d$ to be even. There are 4 choices for $a$, namely $2, 4, 6, 8$. ($a$ cannot equal 0.) There are 5 choices for each of $b, c$ and $d$, namely $0, 2, 4, 6, 8$. The choice of each digit is independent, and so the total number of such integers is $4 \times 5 \times 5 \times 5$ or 500.