In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i, μ_i) (i=1,2) be spaces of finite measure, and let (T, μ) be the product of these spaces. Let E be an ideal in the space S(T₁, μ₁) of measurable functions (i.e., from |e₁|≤|e₂|, e₁∈ S (T₁, μ₁), e₂∈E it follows that e₁∈E). THEOREM 2. Let U be a linear operator from E into S(T₂, μ₂). The following statements are equivalent: 1) there exists a μ-measurable kernel K(t,S) such that (Ue)(S)=∫K(t,S) e(t)d μ(t) (e∈E); 2) if 0≤e_n≤∈E (n=1,2,...) and e_n→0 in measure, then (Ue_n)(S) →0 μ₂ a.e. THEOREM 3. Assume that the function F{cyrillic}(t,S) is such that for any e∈E and for s a.e., the μ₂-measurable function Y(S)=∫F{cyrillic}(t,S)e(t)d μ₁(t) is defined. Then there exists a μ-measurable function K(t,S) such that for any e∈E we have ∫F{cyrillic}(t,S)e(t)d μ₁(t)=∫K(t,S)e(t)d μ₁(t)μ₁a.e.