Home > Measure Theory > Reverse Riesz Type Problem

Reverse Riesz Type Problem

Suppose p \in (1,\infty) and consider q such that 1/p+1/q=1. Take (X,\mathcal{A},\mu) a \sigma-finite measure space. Suppose f:X \to \Bbb{C} is a measurable function such that fg \in L^1(\mu) for all g \in L^q(\mu).

Prove that f \in L^p(\mu).

Solution: Define the following functional \displaystyle \phi : L^q \to \mathbb{R},\ \phi(g)= \int_Xf g d \mu. This is well defined and therefore, by the theorem of representation of linear functionals on L^q we get that there exists a function h \in L^p such that \displaystyle\phi(g)=\int_X hg d \mu, \ \forall g \in L^q. This means that we have $latex \displaystyle \int_X(f-h)gd\mu=0,\ \forall g \in L^q $. Now, take X=\bigcup X_n to be a decomposition of X into countably many measurable spaces with finite measure. For g= g\chi_{X_k} we have that \displaystyle\int_{X_k}(f-h)gd\mu=0,\ \forall g\in L^q(X_k,\mu). Since the space X_k has finite measure, all step functions are contained in L^q(X_k,\mu), and this means that there exists a sequence of step functions (g_n)\subset L^q(X_k,\mu) such that g_n \to f-h, proving that \displaystyle\int_{X_k}(f-h)^2=0 and therefore f=h almost everywhere on X_k. Doing this for all k we see that f=h a.e. on X and therefore f \in L^p(\mu).

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