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This paper begins with new definitions for double sequence spaces. These new definitions are constructed, in general, by combining modulus function and nonnegative four-dimensional matrix. We use these definitions to establish inclusion theorems between various sequence spaces such as: If A = (a m,n,k,l) be a nonnegative four-dimensional matrix such that $$ \mathop {\sup }\limits_{m,n} \sum\limits_{k,l = 0,0}^{\infty ,\infty } {a_{m,n,k,l} < \infty } $$ and let f be a modulus, then ??(A, f) ? ??8(A, f) and ??0(A, f) ? ??8(A, f).