From 2f6a1782b11e2fde3a567f8a0ef4c29e18b3ff3a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Tue, 25 Feb 2025 15:26:02 +0100 Subject: [PATCH] chybejici zavorka --- nahodne-veliciny.tex | 2 +- skripta.pdf | Bin 121089 -> 121103 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/nahodne-veliciny.tex b/nahodne-veliciny.tex index b00986e..e5ada0a 100644 --- a/nahodne-veliciny.tex +++ b/nahodne-veliciny.tex @@ -34,7 +34,7 @@ Máme tedy jakýsi obraz míry $P$ v zobrazení $P_X$ čímž se $(\Omega, \math \begin{proof} Důkaz této věty je poměrně technický, hlavní ideou je ``klasický" postup z teorie míry postupným důkazem nejdříve pro charakteristickou funkci, poté pro jednoduchou měřitelnou (nabývající jen konečně mnoha hodnot), pak pro nezápornou měřitelnou a na závěr pro obecnou měřitelnou funkci. - Nechť $g = \chi_B, B \in \mathcal{M}$. Tedy $g(X(\omega) = 1$ pro $X(\omega) \in B$ (a všude jinde nulová), tedy pro $\omega \in X^{-1}(B)$. Potom máme + Nechť $g = \chi_B, B \in \mathcal{M}$. Tedy $g(X(\omega)) = 1$ pro $X(\omega) \in B$ (a všude jinde nulová), tedy pro $\omega \in X^{-1}(B)$. Potom máme $$ \int_\Omega g(X(\omega) dP(\omega) = \int_{X^{-1}(B)} dP(\omega) = P[X^{-1}(B)]. $$ Pro pravou stranu máme $$ \int_\mathbb{M} g(x) dP_X(x) = \int_B dP_X(x) = P_X(B) = P[X^{-1}(B)].$$ diff --git a/skripta.pdf b/skripta.pdf index 096e1491c9a271fbde81e997437c6601862949ed..ac090d598d026a5f8ee084a693cbb25352e75010 100644 GIT binary patch delta 7273 zcmai(MOf4g!0km!2c$z9L2`hhI|b8_!>hx*gqDGky%^eElkB`KXEzcpBgkjv#8AokeG`7?fleGQ3gT6S zS^Z5Ml!`?DC7KIf?LX80qfiDI_7d`?ID>?QmOn4e#A#OYckQ@i@(3&6C5MK56S1Yp z0$i7*OB%{tVZm2)f4lnn)I3i7?w)HS7oBl|2PMI2h_+sUkDst0n3B1_513)yg8%UG zxxcxpPP>y*)YPcD7uowQl7E`km9Dt8CgA)mI~Qbs@JXr=P5tt1>+}A-4lKzL6TK*n zpFi>Tc;12T#TNT)!(5>^ff052ZcohSwEYxnbs?0(T`*)?syMOFMC}04OHuw%M0I=k zjb7Kl`R^JXob#P-0cnvj6~MPSaR76Wll*A;`2~y_fAmI#y!tTp%G-F&iaWo=!%I;vRB>Q`bH~k0d#hZ@J6jBa}HC4`J59d++Bl`O)4j?K-^Fh~DFe?d* zq!Uhfk0I}HY(YS!xksw_i{;?dM@HDbjmg1&g`Q>$r> z2D(Vr94?$-lpf`_0N$!ZG{kE6X@U^b{Ipr=PNPaCf37^l=_2b%WGg{A7nX!6(ecsV z7z9c;wVe><*?-ZV4+3K-pI)t0CT5TX#ZdO*s~1Yq(L#0$+(Be!@=~^(S$wexo+i^@;kqL|MHKTJrKFK-eLl}ko@$RA!P39}xM2rOR^1Gz7DC~tX*N?Lvk*WlE+ zP0oKXO?!XYgId+Ep@91Gb`&oy_){_7N{=k9Bb99UA85>v(~jBWiX0Z_eWi~mL4w(n z>D2UO!fjI`^hD7mra>DzA0O#Ut|SS z%7Kk2yYPr`O904T*7zzIAA_L=%kW~BB_{3S^Wk=Dn;A|Kh%@FQ|Mlm!7I%lO<$)?h zpfBRsMAGs=`4}V{ZfpXn?=4VM*bbjtX~Q)Sd2Zywx zzihRn`sux#c1ILg@mA6u?*5VEL+aWhnovZLFt#ftS_iFDLRI#`AvxlraTzKz)lF(H z|AQ?ob4=pHHEMZk*qk(IMgu#Yzi;mEpkS{eE=$*`CB_S>Zi4s5)(emPehO9lFXv(p z!}(`kp8!PYZ6YhGDD}Y=r!g`b;Y6ApiJ^O2^6jrUN}6|vTAEXtl@Mi?38pnQip-9= zMFD6bxwZrWyau_`S1-6O~f&ta#cb7WNRCmt-HC>X-RaPnTwN^^P zUo+@_|F+GB56_jDE2FhYB5dWHP6m@JS>8?L*z(J#L|%1Tg}zFeM23{5NAxJ;=Fmpk z!W-2#`teIQEe$B(eyEeOUj7ZM(#vaJHoF?glA&L89(0fej~0* zw9PS*aFjMx%*w@rHA3U8lpiHVRuErrxd8i%Ukc7ZV=%p)a#EZ(TrUZWiLB{TQ&pA6 zs}4QUg1@lg%N1-sxk7%ym-Nuv^y_QJ+$m4oyq*B%`{9@KX9?uBC|AN|A$KF$VFB4} z;sT^D8SS$cZzQZzD>NELaHKQ8lqlCrk%S5wP3{@}uR2qIpe8u?9x&cUwP%mHX9 zzRPhx%vM`;JB%rrb%mO1wXQG(=E8E36;OCftFqqHMj2m)-|O~n-jIWsTTuf=wo*p^ z$t<#QZ}pmKIO8tWWyBr zup&3q@ZeH~JQF?qitMa$TE;c*$9twAi2UkopBTFJTf!z3j|5f|guhq3^liH75<^R2 zrsqxor8F`_)*#*NHPz`^nWECEhM)Af1bOQp<=XF|i;NYe861&0YNlAP$pOb#TavUz zu00QmZl39U!^(%7d#CfhFv1B4;Xx~lKcHbbyLuNQQ%wJ633nCU1td`24e3V*j<3RFf`HUgwrnW_=GWi9Ut|b(ZUp+Y4%7 zDXLY{b>q?%e8kP~H(TzRTRkb(p5+QhPeQ&5h_RwKexuaxdw~3roYf(yB?9`4vURvi zo=j}>ft#wu&HL(~fh~Iop~o1*E-kd!phGM76+`(>Ne|;~_qOi6=ik`})%opr!~-=@*=n7MKM@%-cWT{FSY4Koy~D zV$=wujA*$0i1l9sU8@iy7b-e8D(XulZR?f}nt(fB`lYKCa`!hU1r6waURX%H(edJt zq#p{`xy}}arFMQFDf&bz#|^Q~4`M0pxy^n6;%J z`NXa8DdD5GUJTk2=TNYif%iKb+9=y9rwyw_HJY_FOyfc>l}PE!LZvF7SbX`g9$zBt z17M!PcKCy)1K1!*^JKlx!t)cz(alP|o!ak0Ikx6=QR100@A(r;gLv8R?5Fh1Pn}#I zj--f6MU59XWw@bYngsO(&UxTv{5FrI^vd@TOY*J#>kM~7anKy)tC!w=3XKbx?w!A>*|w!=H3UAiIaX{FkCw zRYyG>`t7Thc5Cke%dc;!_z4~s47H6et?cRyo1R8&JAXA?POp1zR|yQ9*1KnKJl%n8 zov3Qhn$c6Q?(D6jl-+I~ly7?&pX^Q4vp*eGm#_>lp^U7niuiKa&aJrOO$+qq4i71M zUo=DM(uvk}>n^U_hi{k=2UJ=wwtR>Wv5S5Ni6I|0>Dqf|4JJ;)96Sx1-d@`HP#RbeG+dCt|~HsW*@ zyT8p9e+UlUQSw5a7~<&iBq+bjiWjmP#pNnMBnfk!IRug~mOK|Gw@3Qjy7U29%~}X# zJ;hnBhzo2V+I%q4JE8DoKKXERjO{scXlQH5&J}DG8t__cflN0~r-G%{H;R3vri!I( z=)ELv;{1e+>#}_6zvWPLzJgIIkM9brxQ%O&Mw!@kzU4^+L@*qyO|5YdYDSnr{AC06 z#?~naui;JA7kZcqTAAp~+BcxiYwJ3VpoYV&zEx*U;HfO*Ex#!?b%a&LK;0gHZZ(?f zDXVra&CIMv&G$>jL&lP^0Kvd?xnT153r=}@++~$7cBzI}X-~2{v)Pnzznd{ZYN1Mb z#qM0vL0{#ORBS<46IP5%l5DQH$59YfbxxvpKH&0Fzse(4ec0;{#e;y~1(s@cJPT&S zxA)NjZIt&r4$|kozhv}xWAr^=WK-t&>0#+Xb!~NM2t!3btY>iRKikh-_jZNPxk_2L z8@Nke+fvtvnNg&VD+Du}8PSXEJbVo+6=0A7d#~W_W+!)QMv;DAV8Es4B3tQj&k$tWW)jHC)Vt) zUU>=K(TwHK8*`SKzB#A~Nky6~C(F4bT*c5g-+QyhcIR9|OWJ_W^9rORcrK;gz*@g^ z^_6?>4q+`eYkB7mR2g^!hac-+`;E+VJ96D&&SB^r`QS zhl8xigz(vhiel2=Yr7M=1IgvZf$`-+>0kdCQjLi;)75mlbW2?_OKDhSrmJBajSY0`i9|6aCbmV< z`@3`JO-IU=%6Mh)Koo)PnVN)1%QHcS_?)kp($K?6w)!s^3!(dgU3T-l@G+y)&_a~4 zrXKOC8?y^)JeTFp=*oJLKL%ev?&s)v`~;KPo9XOyTP1YG9bZGdo>m zD^2-9huBT)bG44Vgw) z-s~u8GUoII)zrFxVq6SknN_aYgY1tn{gD!u;O=1|0=Y>7kVBl zzXDP)fG5_JY7iap*hS;N)Tg8WNiY~Q6NLEeA$NM|0Tb<}NFVe2g*g;C-uRWZ0NctF zE*97dJZwE~Yss;xgo%>8Ab$?c%pt-U|7 z{bQX&hg^$<2t7MI6Lv#6mng%zN8xhSiAUY`cvOD%_x!V65aF6kZ@i^KoVS8Xu8+6&zLPHoVQb#3UYF33h+a?Ktzt{}faWNPMn- z^jjSsoQs~gvxqI;sz`7?DM_WZLFCg@2=^OjALd`f5bF2E<$DrC)~W7azK)Nk^iE=$ zniC|V&Mi;<8fin~RRi`FQ;rtJ6zpo^lwAPQY$X06*v1y8SjT4OJm02XNZ%ur#C~Dg zoCi_ChcMj^B-)Vw?B7D-S9%tm5q~!}J?lTwDjBOSQV1n{G@99VDjqmX$k^FPxCR26 zHV2>O>wfd=9!qy`AIsMHWQC%C8kMSN>ql_7yZ@dru5vdhDX@+x3%wk)w=Pd+M+4)w#?u zD;#LMz?!qrLq2$zoj3g9Y0CS|n3^rr@CvnOwp}G7_@jgk0f=HJ1_x&N$*=(}e{n#9 z2N}Rr^j`pAI;G~KX4>3Ka*Xj#q;Ft7B6f+2<9;fi+TSN*B3g~H_<|-l{rY_2Cejq$ zFCsn}U1W{fQ)jiZUiy&_6?EKya zcNU30izD`=0RI<<@|dRM_OIv;1IxKwVQEL}cU=9o?1^s%9VEJV-Lq^Z&3>KVEn)9H zx->{eFUor2zjfNmu9S>Y@*RPV+Qt;UQE&5H-TiQ>E{zF2gs7`S>;SP@Vx{ifr!4Z5maJk?;H)#Fj zaN_3|Ai}N#98qqG3HbXDJh+lCzS!3K!?a9rM(x)f$mU5e(y~*s&V4<2z}Ce*$DeO? 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