Intereting Posts

Solving $2x – \sin 2x = \pi/2$ for $0 < x < \pi/2$
Proof of Cauchy Riemann Equations in Polar Coordinates
Accumulation points of accumulation points of accumulation points
Integrating $\sec\theta\tan^2\theta d\,\theta$
Equivalence of three properties of a metric space.
Galois Group of $(x^3-5)(x^2-3)$
Probability of Drawing a Card from a Deck
Upper semi continuous, lower semi continuous
A game with two dice
Evaluating $\int\limits_0^\infty \frac{\log x} {(1+x^2)^2} dx$ with residue theory
Proving Riemann Integrability of a function with countably many discontinuities? (No measure theory)
Why is 'catastrophic cancellation' called so?
Game: two pots with coins
Another integral related to Fresnel integrals
Choosing $\lambda$ to yield sparse solution

I have encountered two definitions of weak convergence in $L^1$:

1) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n\mathrm{1}_A)\rightarrow \mathrm{E}(X\mathrm{1}_A)$ for every measurable set $A$.

2) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n f)\rightarrow \mathrm{E}(X\mathrm{1}f)$ for every (essentially) bounded measurable function $f$.

- Prove for every $n,\;\;$ $\sum\limits_{k=1}^{\infty}\left\lfloor \frac{n}{2^{k}}+\frac{1}{2} \right\rfloor=n $
- Example of a non measurable function!
- Big Rudin 1.40: Open Set is a countable union of closed disks?
- Proving that the terms of the sequence $(nx-\lfloor nx \rfloor)$ is dense in $$.
- Computing $ \int_{0}^{\infty} \frac{1}{(x+1)(x+2)…(x+n)} \mathrm dx $
- Find for which value of the parameter $k$ a function is bijective

my question: are 1) and 2) equivalent?

I see that 2) implies 1) (indicators are bounded), but I have difficulties establishing that 1) implies 2). I tried approximating $f$ by simple functions $f_m$, say, assuming $X_n,X$ are nonnegative for simplicity; the problem: I cannot justify the interchange in the order of taking the limits (first with $n$, and then with $m$). any ideas? I would appreciate any sort of help. many thanks!

- How to investigate the $\limsup$, the $\liminf$, the $\sup$, and especially the $\inf$ of the sequence $(\sqrt{|\sin{n}|})_{n=1}^{\infty}$?
- Semi-partition or pre-partition
- Why do we restrict the definition of Lebesgue Integrability?
- Please show $\int_0^\infty x^{2n} e^{-x^2}\mathrm dx=\frac{(2n)!}{2^{2n}n!}\frac{\sqrt{\pi}}{2}$ without gamma function?
- The Gradient as a Row vs. Column Vector
- Are rotations of $(0,1)$ by $n \arccos(\frac{1}{3})$ dense in the unit circle?
- Is the Ratio of Associative Binary Operations to All Binary Operations on a Set of $n$ Elements Generally Small?
- Writing the roots of a polynomial with varying coefficients as continuous functions?
- Why is it that $\mathscr{F} \ne 2^{\Omega}$?
- Polynomials are dense in $L^2$

Using Vitali-Hahn-Saks theorem or Baire category theorem with $\mathcal F$ endowed with the metric $\rho(A,B)=\mu(A\Delta B)$, we can show for each $\varepsilon>0$, there is $\delta>0$ such that if $\mu(A)\lt \delta$ then $|\mathbb E[X_n\chi_A]|\lt \varepsilon$. Taking $A’:=A\cap \{X_n\leqslant 0\}$ and $A”:=A\cap \{X_n\gt 0\}$, we can see that $\mathbb E[|X_n|\chi_A]\lt\varepsilon$ whenever $\mu(A)\lt\delta$. Indeed, for a fixed $\varepsilon\gt 0$, we define $$F_N:=\bigcap_{n\geqslant N}\left\{A\in\mathcal F,\left|\int_AX_n\mathrm d\mu\right|\leqslant\varepsilon\right\}.$$

Each $F_N$ is closed and $\bigcup_NF_N=\mathcal F$, hence by Baire’s theorem, there is $N_0$, $r_0$ and $A_0\in\mathcal F$ such that $B_\rho(A_0,r_0)\subset F_{N_0}$. Let $B$ such that $\mu(B)\lt r_0$. Since $\mu(A_0\Delta (A_0\cup B))\lt r_0$, $\mu(A_0\Delta (A_0\cap B^c))\lt r_0$ and

$$\int_B X_n\mathrm d\mu=\int_{A_0\cup B}X_n\mathrm d\mu-\int_{A_0\cap B^c}X_n\mathrm d\mu,$$

we have $\left|\int_B X_n\mathrm d\mu\right|\lt \varepsilon$ whenever $n\geqslant N_0$ and $\mu(B)\lt r_0$.

Now we use Theorem 1.12.9 in Bogachev, *Measure theory*, volume 1:

Let $(\Omega,\mathcal F,\mu)$ be a measure space with a finite non-negative measure. Then for each $\delta>0$, we can find an integer $N$ and a finite partition of $\Omega$, $\{S_1,\dots,S_N\}$ such that for each $i$, either $\mu(S_i)\leqslant \delta$ or $S_i$ is an atom of measure $>\delta$.

So take $\varepsilon:=1$, the associated $\delta$, and notice that there are only finitely many atoms of measure $\gt \delta$. On each of these atoms, $X_n$ is constant.

- Convergence of the sequence $\frac{1}{n\sin(n)}$
- A fundamental solution for the Laplacian from a fundamental solution for the heat equation
- Parabola in parametric form
- Improved Betrand's postulate
- In what sense is a tesseract (shown) 4-dimensional?
- Continued Fractions periodicity and convolution.
- Variation of the Kempner series – convergence of series $\sum\frac{1}{n}$ where $9$ is not a digit of $1/n$.
- Why $a^n – b^n$ is divisible by $a-b$?
- How Differential got into calculus
- Combinatorial proof that binomial coefficients are given by alternating sums of squares?
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- Complex Numbers vs. Matrix