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Nonlocal quantum field theory (QFT) of one-component scalar field <inline-formula> <math display="inline"> <semantics> <mi>&#966;</mi> </semantics> </math> </inline-formula> in <i>D</i>-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula> as a functional of external source <i>j</i>, coupling constant <i>g</i> and spatial measure <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mi>&#956;</mi> </mrow> </semantics> </math> </inline-formula> is studied. An expression for GF <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula> in terms of the abstract integral over the primary field <inline-formula> <math display="inline"> <semantics> <mi>&#966;</mi> </semantics> </math> </inline-formula> is given. An expression for GF <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula> in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagator <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>L</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> over the separable HS basis. The classification of functional integration measures <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">D</mi> <mfenced open="[" close="]"> <mi>&#966;</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> is formulated, according to which trivial and two nontrivial versions of GF <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula> are obtained. Nontrivial versions of GF <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula> are expressed in terms of 1-norm and 0-norm, respectively. In the 1-norm case in terms of the original symbol for the product integral, the definition for the functional integration measure <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">D</mi> <mfenced open="[" close="]"> <mi>&#966;</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> over the primary field is suggested. In the 0-norm case, the definition and <i>the meaning</i> of 0-norm are given in terms of the replica-functional Taylor series. The definition of the 0-norm generator <inline-formula> <math display="inline"> <semantics> <mi>&#936;</mi> </semantics> </math> </inline-formula> is suggested. Simple cases of sharp and smooth generators are considered. An alternative derivation of GF <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula> in terms of 0-norm is also given. All these definitions allow to calculate corresponding functional integrals over <inline-formula> <math display="inline"> <semantics> <mi>&#966;</mi> </semantics> </math> </inline-formula> in quadratures. Expressions for GF <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula> in terms of integrals over the separable HS, aka the basis functions representation, with new integrands are obtained. For polynomial theories <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>&#966;</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> and for the nonpolynomial theory <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mo form="prefix">sinh</mo> <mn>4</mn> </msup> <mi>&#966;</mi> </mrow> </semantics> </math> </inline-formula>, integrals over the separable HS in terms of a power series over the inverse coupling constant <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>/</mo> <msqrt> <mi>g</mi> </msqrt> </mrow> </semantics> </math> </inline-formula> for both norms (1-norm and 0-norm) are calculated. Thus, the strong coupling expansion in all theories considered is given. &#8220;Phase transitions&#8221; and critical values of model parameters are found numerically. A generalization of the theory to the case of the uncountable integral over HS is formulated&#8212;GF <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula> for an arbitrary QFT and the strong coupling expansion for the theory <inline-formula> <math display="inline"> <semantics> <msup> <mi>&#966;</mi> <mn>4</mn> </msup> </semantics> </math> </inline-formula> are derived. Finally a comparison of two GFs <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula>, one on the continuous lattice of functions and one obtained using the Parseval&#8722;Plancherel identity, is given.