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The weak L1 -space meets in many areas of mathematics. For example, the conjugate functions of Lebesgue integrable functions belong to the weak L1 -space. The difficulty of working with the weak L1 -space is that the weak L1 -space is not a normed space. Moreover, infinitely differentiable (even continuous) functions are not dense in this space. Due to this, the theory of approximation was not produced in this space. In the present paper, we introduced the concept of the modulus of continuity of the functions from the weak L1 -space, studied its properties, found a criterion for convergence to zero of the modulus of continuity of the function from the weak L1 -space, and proved in this space an analogue of the Jackson-type theorem. [ABSTRACT FROM AUTHOR]