Let $(X,\mathcal{M},\mu)$ be a measure space and $f:X\rightarrow\mathbb{C}$ be a measurable function such that $\int_X |f|\; d\mu <+\infty$. I am trying to prove the following result: \begin{equation} \forall \epsilon >0,\exists \delta >0, \forall E\in\mathcal{M}, \mu(E)<\delta\Rightarrow \int_E...

Hello all mathematicians!! Again, I am struggling with solving the exercises in Lebesgue Integral for preparing the quiz. At this moment, I and my friend are handling this problem, but both of us agreed this problem is a bit tricky. The problem is following. Let ($X,\mathcal{A},\mu$) be a measu...

The problem is Let $(X,\cal M, \mu)$ be a measure space and consider $f\in L^1(X,\cal M, \mu)$. Show that for each $\epsilon > 0$ there exists $\delta > 0$ such that $\int_E {|f|d\mu } < \varepsilon $ for all $E\in \cal M$ with $\mu(E) < \delta$. I can see if $f$ is bounded by some $M$, we...

I am studying for a qualifying exam, and I am struggling with this problem since $f$ is not necessarily integrable. Let $(X,\Sigma, \mu)$ be a measure space and let $$\mathcal{L}(\mu) = \{ \text{ measurable } f \quad| \quad \chi_Ef \in L^1(\mu) \text{ whenever } \mu(E)<\infty\}.$$ Show that fo...