Expected integration approximation under general equal measure partition

In this paper, we first use an $L_2-$discrepancy bound to give the expected uniform integration approximation for functions in the Sobolev space $H^1\left(K\right)$ equipped with a reproducing kernel. The concept of stratified sampling under general equal measure partition is introduced into the research. For different sampling modes, we obtain a better convergence order $O\left(N^{-1-\frac{1}{d}}\right)$ for the stratified sampling set than for the Monte Carlo sampling method and the Latin hypercube sampling method. Second, we give several expected uniform integration approximation bounds for functions equipped with boundary conditions in the general Sobolev space $F_{d,q}^{\ast }$, where $\frac{1}{p}+\frac{1}{q}=1$. Probabilistic $L_p-$discrepancy bound under general equal measure partition, including the case of Hilbert space-filling curve-based sampling are employed. All of these give better general results than simple random sampling, and in particular, Hilbert space-filling curve-based sampling gives better results than simple random sampling for the appropriate sample size.