Bochner Integrable

FunctionalAnalysisAnalysisMeasureTheory Similar to standard Lebesgue integration theory (as well as integration of Progressive Processes), we begin by defining the integral for simple Banach Space Valued Functions: and subsequently for arbitrary Bochner Measurable functions via approximation by simple functions , s.t. : Indeed, we have the following necessary and sufficient conditions for a Bochner Measurable function to be Bochner Integrable:

3. for all

Examples:

3. The Sobolev Space