# Tagged Questions

26 views

### Existence of measurable fuction on non-atomic measure space whose integral is infinity

Let $(X,M,\mu)$ be non atomic measure space with $\mu(X)>0.$ Show that there is a measurable function $f:X\to [0,\infty),$ for which $\int f(x)d\mu(x)=\infty.$ No idea at all. I am preparing for ...
39 views

### Can the theory of Lebesgue integration be extended in a way analogous to extending Riemann integrals to improper Riemann integrals?

I recently (last night) learned the definition of Lebesgue integration and one of the limitations I was told was that some improper Riemann integrals aren't Lebesgue integrable. It occurred to me ...
43 views

### Conditions on a measure

Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>0$ $t^s\in L^1(\mathrm d\mu(t))$, but functions $\mathbf{1}_{t>0}$ and $\mathbf{1}_{t\in (0,1)}$ are not necessarily in ...
190 views

### $\int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=?$

If $f(x) = \frac{1}{(1+x)\sqrt x}$ how to find all $p > 0$ such that $$\int^{\infty}_0 |f(x)|^p dx < \infty$$ The integral is with respect to lebesgue measure. Any solution or hints would ...
103 views

### The relation between arbitrary measure space and the Lebesgue integral

Let $(X, \mathcal F, \mu)$ be a measure space and $f\in M^+(X,\mu)$ (the measurable non-negative functions), and $t>0$. Now let $$S_f(t)=\{x\in X:f(x)>t\} \quad \Psi_f(t)=\mu(S_f(t))$$ Prove ...
136 views

### the integral of a periodic function

Consider $f:\mathbb{R}\to \mathbb{C}$ a bounded and $1$-periodic function, and $g \in L^1(R)$ then $$\lim_{n\to \infty} \int _{R}g(x)f(nx)dx=\int_0^1f(s)ds\int_R g(t)dt.$$ I think the fact that $f$ ...
The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.) Why is it we ...