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For measurable f,g:Rā[0,ā)f,g:\mathbb{R}\to[0,\infty)f,g:Rā[0,ā), their convolution is (fāg)(x)=ā«Rf(t)g(xāt)ādt(f*g)(x)=\int\limits _{\mathbb{R}}f(t)g(x-t) \, dt (fāg)(x)=Rā«āf(t)g(xāt)dt
- f,gāL1ā āā¹ā āfāgāL1f,g\in L^{1}\implies f*g\in L^{1}f,gāL1ā¹fāgāL1 - f,gā„0ā āā¹ā āfāgāL1f,g\ge 0\implies f*g\in L^{1}f,gā„0ā¹fāgāL1 - This allows us to apply Fubini-Tonelliā¦
Fubini-Tonelli
Summary of MATH 895