Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$L$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>L</mi> </math> </EquationSource> </InlineEquation> be a linear space of real random variables on the measurable space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$(\varOmega ,\mathcal {A})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="italic">Ω</mi> <mo>,</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>. Conditions for the existence of a probability <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$P$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>P</mi> </math> </EquationSource> </InlineEquation> on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathcal {A}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="script">A</mi> </math> </EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$E_P|X|\infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>E</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> <mo></mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$E_P(X)=0$$</EquationSource> <EquationSource Format="MATHML">...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>